Methods and systems for lithography calibration

ABSTRACT

A method of efficient optical and resist parameters calibration based on simulating imaging performance of a lithographic process utilized to image a target design having a plurality of features. The method includes the steps of determining a function for generating a simulated image, where the function accounts for process variations associated with the lithographic process; and generating the simulated image utilizing the function, where the simulated image represents the imaging result of the target design for the lithographic process. Systems and methods for calibration of lithographic processes whereby a polynomial fit is calculated for a nominal configuration of the optical system and which can be used to estimate critical dimensions for other configurations.

CROSS-REFERENCE TO RELATED APPLICATIONS

The present application is a divisional of U.S. application Ser. No. 12/613,221, filed Nov. 5, 2009, and issued as U.S. Pat. No. 8,418,088, which claims priority to U.S. Prov. Appln. No. 61/112,130, filed Nov. 6, 2008, the contents of all of which are incorporated by reference herein in their entirety.

FIELD

The invention relates to a method of calibrating a photolithographic system, to a computer readable medium bearing a computer program for calibrating a photolithographic system and a calibration method.

BACKGROUND

Lithographic apparatus can be used, for example, in the manufacture of integrated circuits (ICs). In such a case, the mask may contain a circuit pattern corresponding to an individual layer of the IC, and this pattern can be imaged onto a target portion (e.g. comprising one or more dies) on a substrate (silicon wafer) that has been coated with a layer of radiation-sensitive material (resist). In general, a single wafer will contain a whole network of adjacent target portions that are successively irradiated via the projection system, one at a time. In one type of lithographic projection apparatus, each target portion is irradiated by exposing the entire mask pattern onto the target portion in one go; such an apparatus is commonly referred to as a wafer stepper. In an alternative apparatus, commonly referred to as a step-and-scan apparatus, each target portion is irradiated by progressively scanning the mask pattern under the projection beam in a given reference direction (the “scanning” direction) while synchronously scanning the substrate table parallel or anti-parallel to this direction. Since, in general, the projection system will have a magnification factor M (generally <1), the speed Vat which the substrate table is scanned will be a factor M times that at which the mask table is scanned. More information with regard to lithographic devices as described herein can be gleaned, for example, from U.S. Pat. No. 6,046,792, incorporated herein by reference.

In a manufacturing process using a lithographic projection apparatus, a mask pattern is imaged onto a substrate that is at least partially covered by a layer of radiation-sensitive material (resist). Prior to this imaging step, the substrate may undergo various procedures, such as priming, resist coating and soft baking. After exposure, the substrate may be subjected to other procedures, such as a post-exposure bake (PEB), development, a hard bake and measurement/inspection of the imaged features. This array of procedures is used as a basis to pattern an individual layer of a device, e.g., an IC. Such a patterned layer may then undergo various processes such as etching, ion-implantation (doping), metallization, oxidation, chemo-mechanical polishing, etc., all intended to finish off an individual layer. If several layers are required, then the whole procedure, or a variant thereof, will have to be repeated for each new layer. Eventually, an array of devices will be present on the substrate (wafer). These devices are then separated from one another by a technique such as dicing or sawing, whence the individual devices can be mounted on a carrier, connected to pins, etc.

For the sake of simplicity, the projection system may hereinafter be referred to as the “lens”; however, this term should be broadly interpreted as encompassing various types of projection systems, including refractive optics, reflective optics, and catadioptric systems, for example. The radiation system may also include components operating according to any of these design types for directing, shaping or controlling the projection beam of radiation, and such components may also be referred to below, collectively or singularly, as a “lens”. Further, the lithographic apparatus may be of a type having two or more substrate tables (and/or two or more mask tables). In such “multiple stage” devices the additional tables may be used in parallel, or preparatory steps may be carried out on one or more tables while one or more other tables are being used for exposures. Twin stage lithographic apparatus are described, for example, in U.S. Pat. No. 5,969,441, incorporated herein by reference.

The photolithographic masks referred to above comprise geometric patterns corresponding to the circuit components to be integrated onto a silicon wafer. The patterns used to create such masks are generated utilizing CAD (computer-aided design) programs, this process often being referred to as EDA (electronic design automation). Most CAD programs follow a set of predetermined design rules in order to create functional masks. These rules are set by processing and design limitations. For example, design rules define the space tolerance between circuit devices (such as gates, capacitors, etc.) or interconnect lines, so as to ensure that the circuit devices or lines do not interact with one another in an undesirable way. The design rule limitations are typically referred to as “critical dimensions” (CD). A critical dimension of a circuit can be defined as the smallest width of a line or hole or the smallest space between two lines or two holes. Thus, the CD determines the overall size and density of the designed circuit. Of course, one of the goals in integrated circuit fabrication is to faithfully reproduce the original circuit design on the wafer (via the mask).

As noted, microlithography is a central step in the manufacturing of semiconductor integrated circuits, where patterns formed on semiconductor wafer substrates define the functional elements of semiconductor devices, such as microprocessors, memory chips etc. Similar lithographic techniques are also used in the formation of flat panel displays, micro-electro mechanical systems (MEMS) and other devices.

As semiconductor manufacturing processes continue to advance, the dimensions of circuit elements have continually been reduced while the amount of functional elements, such as transistors, per device has been steadily increasing over decades, following a trend commonly referred to as ‘Moore's law’. At the current state of technology, critical layers of leading-edge devices are manufactured using optical lithographic projection systems known as scanners that project a mask image onto a substrate using illumination from a deep-ultraviolet laser light source, creating individual circuit features having dimensions well below 100 nm, i.e. less than half the wavelength of the projection light.

This process, in which features with dimensions smaller than the classical resolution limit of an optical projection system are printed, is commonly known as low-k₁ lithography, according to the resolution formula CD=k₁×λ/NA, where λ is the wavelength of radiation employed (currently in most cases 248 nm or 193 nm), NA is the numerical aperture of the projection optics, CD is the ‘critical dimension’—generally the smallest feature size printed—and k₁ is an empirical resolution factor. In general, the smaller k₁, the more difficult it becomes to reproduce a pattern on the wafer that resembles the shape and dimensions planned by a circuit designer in order to achieve particular electrical functionality and performance. To overcome these difficulties, sophisticated fine-tuning steps are applied to the projection system as well as to the mask design. These include, for example, but not limited to, optimization of NA and optical coherence settings, customized illumination schemes, use of phase shifting masks, optical proximity correction in the mask layout, or other methods generally defined as ‘resolution enhancement techniques’ (RET).

As one important example, optical proximity correction (OPC, sometimes also referred to as ‘optical and process correction’) addresses the fact that the final size and placement of a printed feature on the wafer will not simply be a function of the size and placement of the corresponding feature on the mask. It is noted that the terms ‘mask’ and ‘reticle’ are utilized interchangeably herein. For the small feature sizes and high feature densities present on typical circuit designs, the position of a particular edge of a given feature will be influenced to a certain extent by the presence or absence of other adjacent features. These proximity effects arise from minute amounts of light coupled from one feature to another. Similarly, proximity effects may arise from diffusion and other chemical effects during post-exposure bake (PEB), resist development, and etching that generally follow lithographic exposure.

In order to ensure that the features are generated on a semiconductor substrate in accordance with the requirements of the given target circuit design, proximity effects need to be predicted utilizing sophisticated numerical models, and corrections or pre-distortions need to be applied to the design of the mask before successful manufacturing of high-end devices becomes possible. The article “Full-Chip Lithography Simulation and Design Analysis—How OPC Is Changing IC Design”, C. Spence, Proc. SPIE, Vol. 5751, pp 1-14 (2005) provides an overview of current ‘model-based’ optical proximity correction processes. In a typical high-end design almost every feature edge requires some modification in order to achieve printed patterns that come sufficiently close to the target design. These modifications may include shifting or biasing of edge positions or line widths as well as application of ‘assist’ features that are not intended to print themselves, but will affect the properties of an associated primary feature.

The application of model-based OPC to a target design requires good process models and considerable computational resources, given the many millions of features typically present in a chip design. However, applying OPC is generally not an ‘exact science’, but an empirical, iterative process that does not always resolve all possible weaknesses on a layout. Therefore, post-OPC designs, i.e. mask layouts after application of all pattern modifications by OPC and any other RETs, need to be verified by design inspection, i.e. intensive full-chip simulation using calibrated numerical process models, in order to minimize the possibility of design flaws being built into the manufacturing of a mask set. This is driven by the enormous cost of making high-end mask sets, which run in the multi-million dollar range, as well as by the impact on turn-around time by reworking or repairing actual masks once they have been manufactured.

Both OPC and full-chip RET verification may be based on numerical modeling systems and methods as described, for example in, U.S. Pat. No. 7,003,758 and an article titled “Optimized Hardware and Software For Fast, Full Chip Simulation”, by Y. Cao et al., Proc. SPIE, Vol. 5754, 405 (2005).

While full-chip numerical simulation of the lithographic patterning process has been demonstrated at a single process condition, typically best focus and best exposure dose or best ‘nominal’ condition, it is well known that manufacturability of a design requires sufficient tolerance of pattern fidelity against small variations in process conditions that are unavoidable during actual manufacturing. This tolerance is commonly expressed as a process window, defined as the width and height (or ‘latitude’) in exposure-defocus space over which CD or edge placement variations are within a predefined margin (i.e., error tolerance), for example ±10% of the nominal line width. In practice, the actual margin requirement may differ for different feature types, depending on their function and criticality. Furthermore, the process window concept can be extended to other basis parameters in addition to or besides exposure dose and defocus.

Manufacturability of a given design generally depends on the common process window of all features in a single layer. While state-of-the-art OPC application and design inspection methods are capable of optimizing and verifying a design at nominal conditions, it has been recently observed that process-window aware OPC models will be required in order to ensure manufacturability at future process nodes due to ever-decreasing tolerances and CD requirements.

Currently, in order to map out the process window of a given design with sufficient accuracy and coverage, simulations at N parameter settings (e.g., defocus and exposure dose) are required, where N can be on the order of a dozen or more. Consequently, an N-fold multiplication of computation time is necessary if these repeated simulations at various settings are directly incorporated into the framework of an OPC application and verification flow, which typically will involve a number of iterations of full-chip lithography simulations. However, such an increase in the computational time is prohibitive when attempting to validate and/or design a given target circuit.

As such, there is a need for simulation methods and systems which account for variations in the process-window that can be used for OPC and RET verification, and that are more computationally efficient than such a ‘brute-force’ approach of repeated simulation at various conditions as is currently performed by known prior art systems.

SUMMARY

According to certain aspects of the invention, there is provided a calibration method which allows for a computation efficient technique and which overcomes the foregoing deficiencies of the prior art techniques. More specifically, certain embodiments of the present invention relate to a method of optical and resist parameters calibration based on simulating imaging performance of a lithographic process utilized to image a target design having a plurality of features and provide an efficient calibration methodology based on polynomial expansion of aerial and resist images.

Methods according to certain aspects of the invention include the steps of determining a function for generating a simulated image, where the function accounts for process variations associated with the lithographic process; and generating the simulated image utilizing the function, where the simulated image represents the imaging result of the target design for the lithographic process. In one given embodiment, the function is defined as:

I(x,f)=I ₀(x)+a(x)(f−f ₀)+b(x)(f−f ₀)²

where I_(O) represents image intensity at nominal focus, f_(O) represents nominal focus, f represents an actual focus level at which the simulated image is calculated, and parameters “a” and “b” represent first order and second order derivative images.

According to a further aspect of the invention there is provided a calibration method, comprising generating a model for a lithographic process, the model comprising a polynomial series expansion around a nominal value of a physical parameter of the lithographic process. The calibration method further comprises fitting the model to measured dimensions of imaging results obtained by applying the lithographic process at a plurality of values of the physical parameter by optimizing at least one polynomial expansion coefficient. Note that by applying a polynomial series expansion, a separation of the model into an optical model part and a resist model part is not required: the polynomial series expansion can be made for the integrated (i.e. combined) model. Additionally the calibration is reduces to a linear problem even if the parameters are non-linear.

The present invention provides significant advantages over prior art methods. Most importantly, the present invention provides a computational efficient simulation process with accounts for variations in the process window (e.g., focus variations and exposure dose variations), and eliminates the need to perform the ‘brute-force’ approach of repeated simulation at various conditions as is currently practiced by known prior art methods.

Additionally it is to be noted that calibration procedures for lithography models are required that provide models being valid, robust and accurate across the process window, not only at singular, specific parameter settings. Prior art calibration procedures, typically start from certain number of known mask patterns with corresponding wafer measurements (such as wafer CDs or contours). Then the model parameters (both optical and resist) are determined so that the predictions (CDs, contours, etc) from the model match the actual measurements. Since the relationship between the model prediction and model parameters is quite complicated, known prior art systems rely on brute-force search in calibration, especially for optical parameter calibration. In such brute-force search, users have to first identify a search range for each model parameter. Then for each possible parameter values combination (i.e., possible process condition), CDs or contours are predicted from all known mask patterns and compared against the wafer measurements. After exhausting all possible parameter values combinations, the one which results in best match between predictions and measurements is chosen. This brute-force approach has two principal drawbacks: (1) it is time consuming because model construction from each parameter values combination is computationally expensive, and (2) it is difficult to determine the search range. If the search range is too large, the calibration process may be prohibitively slow; while if the search range is too small, it may not contain the right parameter values combination at all.

This problem is solved by an aspect of the invention providing a calibration method, comprising a further polynomial series expansion around a further nominal value of a further physical parameter of the lithographic process, and wherein the imaging results are obtained at a plurality of values of the further physical parameter and wherein fitting the model comprises optimizing at least one further polynomial expansion coefficient corresponding to the further polynomial series expansion. By using two polynomial series expansions for the physical parameter and the further physical parameter, the equations to be solved are linearised enabling an efficient way to solve them. Because of this use of two polynomial series expansions, the fits to the model do not rely on the combinations of the values of the physical parameters (i.e. do not pose restrictions to the combinations) used for applying the lithographic process.

Although specific reference may be made in this text to the use of the invention in the manufacture of ICs, it should be explicitly understood that the invention has many other possible applications. For example, it may be employed in the manufacture of integrated optical systems, guidance and detection patterns for magnetic domain memories, liquid-crystal display panels, thin-film magnetic heads, etc. The skilled artisan will appreciate that, in the context of such alternative applications, any use of the terms “reticle”, “wafer” or “die” in this text should be considered as being replaced by the more general terms “mask”, “substrate” and “target portion”, respectively.

In the present document, the terms “radiation” and “beam” are used to encompass all types of electromagnetic radiation, including ultraviolet radiation (e.g. with a wavelength of 365, 248, 193, 157 or 126 nm) and EUV (extreme ultra-violet radiation, e.g. having a wavelength in the range 5-20 nm).

The term “mask” as employed in this text may be broadly interpreted as referring to generic patterning means that can be used to endow an incoming radiation beam with a patterned cross-section, corresponding to a pattern that is to be created in a target portion of the substrate; the term “light valve” can also be used in this context. Besides the classic mask (transmissive or reflective; binary, phase-shifting, hybrid, etc.), examples of other such patterning means include:

-   -   a programmable mirror array. An example of such a device is a         matrix-addressable surface having a viscoelastic control layer         and a reflective surface. The basic principle behind such an         apparatus is that (for example) addressed areas of the         reflective surface reflect incident light as diffracted light,         whereas unaddressed areas reflect incident light as undiffracted         light. Using an appropriate filter, the said undiffracted light         can be filtered out of the reflected beam, leaving only the         diffracted light behind; in this manner, the beam becomes         patterned according to the addressing pattern of the         matrix-addressable surface. The required matrix addressing can         be performed using suitable electronic means. More information         on such mirror arrays can be gleaned, for example, from U.S.         Pat. No. 5,296,891 and U.S. Pat. No. 5,523,193, which are         incorporated herein by reference.     -   a programmable LCD array. An example of such a construction is         given in U.S. Pat. No. 5,229,872, which is incorporated herein         by reference.

The invention itself, together with further objects and advantages, can be better understood by reference to the following detailed description and the accompanying schematic drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

Embodiments of the invention will now be described, by way of example only, with reference to the accompanying schematic drawings in which corresponding reference symbols indicate corresponding parts, and in which:

FIG. 1 depicts block diagram illustrating a typical lithographic projection system;

FIG. 2 depicts functional modules of a lithographic simulation model in accordance with one embodiment of the invention;

FIG. 3 depicts flowchart in accordance with one embodiment of the invention;

FIG. 4 depicts a flowchart in accordance with one embodiment of the invention;

FIG. 5 depicts a flowchart in accordance with one embodiment of the invention;

FIG. 6 depicts a computer system employed in one embodiment of the invention;

FIG. 7 depicts a lithographic projection apparatus in accordance with one embodiment of the invention;

FIG. 8 depicts a prior art lithography calibration process; and

FIG. 9 depicts a flowchart in accordance with one embodiment of the invention;

FIG. 10 depicts a flowchart in accordance with one embodiment of the invention.

DETAILED DESCRIPTION

FIG. 1 illustrates one example of a lithographic projection system 10. The major components include a light source 12, which may be a deep-ultraviolet excimer laser source, illumination optics which define the partial coherence (denoted as sigma) and which may include specific source shaping optics 14, 16 a and 16 b; a mask or reticle 18; and projection optics 16 c that produce an image of the reticle pattern on wafer plane 22. An adjustable filter or aperture 20 at the pupil plane may restrict the range of beam angles that impinge on the wafer plane 22, where the largest possible angle defines the numerical aperture of the projection optics NA=sin(θ_(max)).

In a lithography simulation system, these major system components can be described by separate functional modules as illustrated in the example of FIG. 2. Referring to FIG. 2, functional modules include the design layout module 26, which defines the target design; the mask layout module 28, which defines the mask to be utilized in imaging process, the mask model module 30, which defines the model of the mask layout to be utilized during the simulation process, the optical model module 32, which defines the performance of the optical components of lithography system and the resist model module 34, which defines the performance of the resist being utilized in the given process. As is known, the result of the simulation process produces, for example, predicted contours and CDs in the result module 36.

The properties of the illumination and projection optics are typically captured in optical model 32 that includes NA-sigma (σ) settings as well as any particular illumination source shape. The optical properties of the photo-resist layer coated on a substrate—i.e. refractive index, film thickness, propagation and polarization effects—may also be captured as part of the optical model 32. The mask model 30 captures the design features of the reticle and may also include a representation of detailed physical properties of the mask, as described, for example, in U.S. Patent Application No. 60/719,837. The resist model 34 describes the effects of chemical processes which occur during resist exposure, PEB and development, in order to predict, for example, contours of resist features formed on the substrate wafer. An objective of the simulation is to accurately predict edge placements, CDs, etc., which can then be compared against a target design. The target design, is generally defined as a pre-OPC mask layout, and will typically be provided in a standardized digital file format such as GDSII or OASIS.

In certain embodiments, the connection between the optical and the resist model is a simulated aerial image within the resist layer, which arises from the projection of light onto the substrate, refraction at the resist interface and multiple reflections in the resist film stack. The light intensity distribution (“aerial image”) is turned into a latent resist image by absorption of photons, which is further modified by diffusion processes and various loading effects. Efficient simulation methods that are fast enough for full-chip applications approximate the realistic 3-dimensional intensity distribution in the resist stack by a 2-dimensional aerial and/or resist image. An efficient implementation of a lithography model is possible using the following formalism in which the image is expressed as a Fourier sum over signal amplitudes in the pupil plane. The image is expressed here in a scalar form which may be extended to include polarization vector effects. According to the known Hopkins theory, the aerial image may be defined by:

$\begin{matrix} \begin{matrix} {{I(x)} = \left. \Sigma_{k} \middle| {{A(k)}\Sigma_{k^{\prime}}{M\left( {k^{\prime} - k} \right)}{P\left( k^{\prime} \right)}{\exp \left( {{- j}\; k^{\prime}x} \right)}} \right|^{2}} \\ {= {\Sigma_{k}{A(k)}^{2}\left\{ {\Sigma_{k^{\prime}}\Sigma_{k^{''}}{M\left( {k^{\prime} - k} \right)}{P\left( k^{\prime} \right)}{M^{*}\left( {k^{''} - k} \right)}{P^{*}\left( k^{''} \right)}{\exp \left( {{- {j\left( {k^{\prime} - k^{''}} \right)}}x} \right)}} \right\}}} \\ {= {\Sigma_{k^{\prime}}{\Sigma_{k^{''}}\left\lbrack {\Sigma_{k}{A(k)}^{2}{P\left( {k + k^{\prime}} \right)}{P^{*}\left( {k + k^{''}} \right)}} \right\rbrack}{M\left( k^{\prime} \right)}{M^{*}\left( k^{''} \right)}{\exp \left( {{- {j\left( {k^{\prime} - k^{''}} \right)}}x} \right)}}} \\ {= {\Sigma_{k^{\prime}}\Sigma_{k^{''}}{TCC}_{k^{\prime},k^{''}}{M\left( k^{\prime} \right)}{M^{*}\left( k^{''} \right)}{\exp \left( {{- {j\left( {k^{\prime} - k^{''}} \right)}}x} \right)}}} \end{matrix} & \left( {{Eq}.\mspace{14mu} 1} \right) \end{matrix}$

where, I (x) is the aerial image intensity at point x within the image plane (for notational simplicity, a two-dimensional coordinate represented by a single variable is utilized), k represents a point on the source plane, A (k) is the source amplitude from point k, k′ and k″ are points on the pupil plane, M is the Fourier transform of the mask image, P is the pupil function, and TCC_(k′,k″)=Σ_(k)A(k)² P(k+k′)P*(k+k″). A notable aspect of the foregoing derivation is the change of summation order (moving the sum over k inside) and indices (replacing k′ with k+k′ and replacing k″ with k+k′), which results in the separation of the Transmission Cross Coefficients (TCCs), defined by the term inside the square brackets in the third line in the equation. These coefficients are independent of the mask pattern and therefore can be pre-computed using knowledge of the optical elements or configuration only (e.g., NA and a or the detailed illuminator profile). It is further noted that although in the given example (Eq.1) is derived from a scalar imaging model, this formalism can also be extended to a vector imaging model, where TE and TM polarized light components are summed separately.

Furthermore, the approximate aerial image can be calculated by using only a limited number of dominant TCC terms, which can be determined by diagonalizing the TCC matrix and retaining the terms corresponding to its largest eigenvalues, i.e.,

$\begin{matrix} {{TCC}_{k^{\prime},k^{''}} = {\sum\limits_{i = 1}^{N}\; {\lambda_{i}{\varphi_{i}\left( k^{\prime} \right)}{\varphi_{i}^{*}\left( k^{''} \right)}}}} & \left( {{Eq}.\mspace{14mu} 2} \right) \end{matrix}$

where λ_(i)(i=1, . . . , N) denotes the N largest eigenvalues and φ_(i)() denotes the corresponding eigenvector of the TCC matrix. It is noted that (Eq.2) is exact when all terms are retained in the Eigen series expansion, i.e., when N is equal to the rank of the TCC matrix. However, in actual applications, it is typical to truncate the series by selecting a smaller N to increase the speed of the computation process. Thus, (Eq.1) can be rewritten as:

$\begin{matrix} {\begin{matrix} {{I(x)} = {\underset{k^{\prime}}{\Sigma}\underset{k^{''}}{\Sigma}{TCC}_{k^{\prime},k^{''}}{M\left( k^{\prime} \right)}{M^{*}\left( k^{''} \right)}{\exp \left( {{- {j\left( {k^{\prime} - k^{''}} \right)}}x} \right)}}} \\ {= {\underset{k^{\prime}}{\Sigma}\underset{k^{''}}{\Sigma}{\sum\limits_{i = 1}^{N}\; {\lambda_{i}{\varphi_{i}\left( k^{\prime} \right)}{\varphi_{i}^{*}\left( k^{''} \right)}{M\left( k^{\prime} \right)}{M^{*}\left( k^{''} \right)}{\exp \left( {{- {j\left( {k^{\prime} - k^{''}} \right)}}x} \right)}}}}} \\ {= {\sum\limits_{i = 1}^{N}\; {\lambda_{i}\underset{k^{\prime}}{\Sigma}{\varphi_{i}\left( k^{\prime} \right)}{M\left( k^{\prime} \right)}{\exp \left( {{- {jk}^{\prime}}x} \right)}\underset{k^{''}}{\Sigma}{\varphi_{i}^{*}\left( k^{''} \right)}{M^{*}\left( k^{''} \right)}{\exp \left( {{jk}^{''}x} \right)}}}} \\ {= \left. {\sum\limits_{i = 1}^{N}\; \lambda_{i}} \middle| {\Phi_{i}(x)} \right|^{2}} \end{matrix}{{{where}\mspace{14mu} {\Phi_{i}(x)}} = {\underset{k^{''}}{\Sigma}{\varphi_{i}\left( k^{''} \right)}{M\left( k^{''} \right)}{\exp \left( {{- {jk}^{''}}x} \right)}}}} & \left( {{Eq}.\mspace{14mu} 3} \right) \end{matrix}$

and || denotes the magnitude of a complex number.

Using a sufficiently large number of TCC terms and a suitable model calibration methodology allows for an accurate description of the optical projection process and provides for ‘separability’ of the lithographic simulation model into the optics and resist models or parts. In an ideal, separable model, all optical effects such as NA, sigma, defocus, aberrations etc. are accurately captured in the optical model module, while only resist effects are simulated by the resist model. In practice, however, all ‘efficient’ lithographic simulation models (as opposed to first-principle models, which are generally too slow and require too many adjustable parameters to be practical for full-chip simulations) are empirical to some extent and tend to use a limited set of parameters. In some cases, “lumped” parameters can account for certain combined net effects of both optical and resist properties. For example, diffusion processes during PEB of resist can be modeled by a Gaussian filter that blurs the image formed in resist, while a similar filter might also describe the effect of stray light, stage vibration, or the combined effect of high-order aberrations of the projection system. Lumped parameters can reproduce process behavior close to fitted calibration points, but typically have inferior predictive power compared with separable models. Separability typically requires a sufficiently detailed model form—for example, the example discussed above may use two independent filters for optical blurring and resist diffusion—as well as a suitable calibration methodology that assures isolation of optical effects from resist effects.

While a separable model may be suited for most applications, the description of through-process window “PW” aerial image variations associated with the method according to certain aspects of the present invention, as set forth in more detail below, does not require strict model separability. Methods for adapting a general resist model in order to accurately capture through-PW variations are also described below.

Certain embodiments provide efficient simulation of lithographic patterning performance covering parameter variations throughout a process window, i.e., a variation of exposure dose and defocus or additional process parameters. Using an image-based approach, certain embodiments provide polynomial series expansions for aerial images or resist images as a function of focus and exposure dose variations, or other additional coordinates of a generalized PW. These expressions involve images and derivative images which relate to TCCs and derivative TCC matrices. Linear combinations of these expressions allow for a highly efficient evaluation of the image generated at any arbitrary PW point. In addition, edge placement shifts or CD variations throughout the PW are also expressed in analytical form as simple linear combinations of a limited set of simulated images. This set of images may be generated within a computation time on the order of approximately 2 times the computation time for computing a single image at NC (Nominal Condition), rather than N× by computing images at N separate PW conditions. Once this set of images is known, the complete through-PW behavior of every single edge or CD on the design can be immediately determined.

Certain embodiments of the invention provide methods that may also be utilized in conjunction with model calibration, lithography design inspection, yield estimates based on evaluating the common PW, identification of hot spots, modification and repair of hot spots by PW-aware OPC, and model-based process control corrections, e.g., to center the common PW of a litho layer.

Certain aspects of the method may be best appreciated understood by considering the example of through-focus changes in resist line width (or edge placement) of a generic resist line. It is well known that the CD of the resist line typically has a maximum or minimum value at best focus, but the CD varies smoothly with defocus in either direction. Therefore, the through-focus CD variations of a particular feature may be approximated by a polynomial fit of CD vs. defocus, e.g. a second-order fit for a sufficiently small defocus range. However, the direction and magnitude of change in CD will depend strongly on the resist threshold (dose to clear), the specific exposure dose, feature type, and proximity effects. Thus, exposure dose and through-focus CD changes are strongly coupled in a non-linear manner that prevents a direct, general parameterization of CD or edge placement changes throughout the PW space.

However, the aerial image is also expected to show a continuous variation through focus. Every mask point may be imaged to a finite-sized spot in the image plane that is characterized by the point spread function of the projection system. This spot will assume a minimum size at best focus but will continuously blur into a wider distribution with both positive and negative defocus. Therefore, it is possible to approximate the variation of image intensities through focus as a second-order polynomial for each individual image point within the exposure field:

I(x,f)=I ₀(x)+a(x)·(f−f ₀)+b(x)·(f−f ₀)²  (Eq.4)

where f₀ indicates the nominal or best focus position, and f is the actual focus level at which the image I is calculated. The second-order approximation is expected to hold well for a sufficiently small defocus range, but the accuracy of the approximation may easily be improved by including higher-order terms if required (for example, 3^(rd) order and/or 4^(th) order terms). In fact, (Eq.4) can also be identified as the beginning terms of a Taylor series expansion of the aerial image around the nominal best focus plane:

$\begin{matrix} {{I\left( {x,f} \right)} = \left. {{I\left( {x,f_{0}} \right)} + \frac{\partial{I\left( {x,f} \right)}}{\partial f}} \middle| {}_{f = f_{0}}{{\cdot \left( {f - f_{0}} \right)} + {2\frac{\partial^{2}{I\left( {x,f} \right)}}{\partial f^{2}}}} \middle| {}_{f = f_{0}}{\cdot \left( {f - f_{0}} \right)^{2}} \right.} & \left( {{Eq}.\mspace{14mu} 5} \right) \end{matrix}$

which can in principle be extended to an arbitrarily sufficient representation of the actual through-focus behavior of the aerial image by extension to include additional higher-order terms. It is noted that the choice of polynomial base functions is only one possibility to express a series expansion of the aerial image through focus, and the methods of the current invention are by no means restricted to this embodiment, e.g., the base functions can be special functions such as Bessel Functions, Legendre Functions, Chebyshev Functions, Trigonometric functions, and so on. In addition, while the process window term is most commonly understood as spanning variations over defocus and exposure dose, the process window concept can be generalized and extended to cover additional or alternative parameter variations, such as variation of NA and sigma, etc.

Comparison of (Eq.4) and (Eq.5) reveals the physical meaning of the parameters “a” and “b” as first and second-order derivative images. These may in principle be determined directly as derivatives by a finite difference method for every image point and entered into (Eq.4) and (Eq.5) to interpolate the image variations. Alternatively, in order to improve the overall agreement between the interpolation and the actual through focus variation over a wider range, the parameters a and b can be obtained from a least square fit of (Eq.4) over a number of focus positions {f₁, f₂, . . . , f_(L)} for which aerial images are explicitly calculated as {I₁, I₂, . . . , I_(L)}. The parameters “a” and “b” are then found as solutions to the following system of equations in a least square sense (assuming here that L>3, in which case the system of equations is over-determined).

Without loss of generality, it is assumed that f₀=0 so as to simplify the notation. Then for a fixed image point,

I ₁ =I ₀ +a·f ₁ +b·f ₁ ²

I ₂ =I ₀ +a·f ₂ +b·f ₂ ²

. . .

I _(L) =I ₀ +a·f _(L) +b·f _(L) ²  (Eq.6)

where I₀ is the aerial image at nominal conditions (NC), i.e. f=f₀. The solution to the above set of equations minimizes the following sum of squared differences, with the index I referring to the L different focus conditions:

$\begin{matrix} {G = {\sum\limits_{l = 1}^{L}\; {W_{l} \cdot \left( {I_{l} - I_{0} - {a \cdot f_{l}} - {b \cdot f_{l}^{2}}} \right)^{2}}}} & \left( {{Eq}.\mspace{14mu} 7} \right) \end{matrix}$

where W_(l) is a user-assigned weight to defocus f_(l)(l=1, 2, . . . , L). Through {W₁, W₂, . . . , W_(L)}, it is possible to assign different weights to different focuses. For example, in order to make the 2^(nd) order polynomial approximation have a better match at PW points closer to NC, it is possible to assign a larger weight close to NC and a smaller weight away from NC; or if it is desired for all focus points to have equal importance, one can simply assign equal weights, i.e., W₁=W₂= . . . =W_(L)=1. For large deviations in focus and dose relative to the nominal condition, many patterns become unstable in printing and the measurements of CDs become unreliable, in such cases it may be desirable to assign small weights to such process window conditions.

To solve (Eq.7), it is noted that the best fit will fulfill the conditions:

$\begin{matrix} {{\frac{\partial G}{\partial a} \equiv 0},{{{and}\mspace{14mu} \frac{\partial G}{\partial b}} \equiv 0}} & \left( {{Eq}.\mspace{14mu} 8} \right) \end{matrix}$

(Eq.8) can be solved analytically, resulting in immediate expressions for “a” and “b” as the linear combination or weighted sum of the {I_(l)}, as shown below. The coefficients of this linear combination do not depend on the pixel coordinate or pattern, but only on the values of the {f_(l)} and {W_(l)}. As such, these coefficients can be understood as forming a linear filter for the purpose of interpolation in the space of f, and the particular choice of polynomials as base functions gives rise to the specific values of the coefficients, independent of the mask pattern. More specifically, the calculation of these coefficients is performed once the values of {f_(l)} and {W_(l)} are determined, without knowing the specific optical exposure settings or actually carrying out aerial image simulations.

With regard to solving (Eq.8), (Eq.7) can be rewritten as:

$\begin{matrix} {G = {\sum\limits_{l = 1}^{L}\; {W_{l} \cdot \left( {I_{l} - I_{0} - {a \cdot f_{l}} - {b \cdot f_{l}^{2}}} \right)^{2}}}} \\ {= {\sum\limits_{l = 1}^{L}\; {W_{l} \cdot \left( {{b \cdot f_{l}^{2}} + {a \cdot f_{l}} - {\Delta \; I_{l}}} \right)^{2}}}} \end{matrix}$

where ΔI_(l)=I_(l)−I₀ for l=1, 2, K, L. As a result, (Eq.8) can be expanded as:

$\begin{matrix} {\frac{\partial G}{\partial a} = {\sum\limits_{l = 1}^{L}\; {{W_{l} \cdot 2}{\left( {{b \cdot f_{l}^{2}} + {a \cdot f_{l}} - {\Delta \; I_{l}}} \right) \cdot f_{l}}}}} \\ {{= {{2{a \cdot {\sum\limits_{l = 1}^{L}\; {W_{l} \cdot f_{l}^{2}}}}} + {2{b \cdot {\sum\limits_{l = 1}^{L}\; {W_{l} \cdot f_{l}^{3}}}}} - {2 \cdot {\sum\limits_{l = 1}^{L}\; {{W_{l} \cdot \Delta}\; {I_{l} \cdot f_{l}}}}}}}} \\ {{= {{2{a \cdot \alpha_{2}}} + {2{b \cdot \alpha_{3}}} - {2\; \Phi_{1}}}}} \\ {{\equiv 0}} \end{matrix}$ $\begin{matrix} {\frac{\partial G}{\partial b} = {\sum\limits_{l = 1}^{L}\; {{W_{l} \cdot 2}{\left( {{b \cdot f_{l}^{2}} + {a \cdot f_{l}} - {\Delta \; I_{l}}} \right) \cdot f_{l}^{2}}}}} \\ {{= {{2{a \cdot {\sum\limits_{l = 1}^{L}\; {W_{l} \cdot f_{l}^{3}}}}} + {2{b \cdot {\sum\limits_{l = 1}^{L}\; {W_{l} \cdot f_{l}^{4}}}}} - {2 \cdot {\sum\limits_{l = 1}^{L}\; {{W_{l} \cdot \Delta}\; {I_{l} \cdot f_{l}^{2}}}}}}}} \\ {{= {{2{a \cdot \alpha_{3}}} + {2{b \cdot \alpha_{4}}} - {2\; \Phi_{2}}}}} \\ {{\equiv 0}} \end{matrix}$

Thus:

$\begin{matrix} {{{a = {\frac{{\alpha_{4}\Phi_{1}} - {\alpha_{3}\Phi_{2}}}{{\alpha_{2}\alpha_{4}} - \alpha_{3}^{2}} = {{\sum\limits_{l = 1}^{L}\; {h_{al}\Delta \; I_{l}}} = {\sum\limits_{l = 1}^{L}\; {h_{al}\left( {I_{l} - I_{0}} \right)}}}}},{b = {\frac{{\alpha_{2}\Phi_{2}} - {\alpha_{3}\Phi_{1}}}{{\alpha_{2}\alpha_{4}} - \alpha_{3}^{2}} = {{\sum\limits_{l = 1}^{L}\; {h_{bl}\Delta \; I_{l}}} = {\sum\limits_{l = 1}^{L}\; {h_{bl}\left( {I_{l} - I_{0}} \right)}}}}},{where}}{{\alpha_{2} = {\sum\limits_{l = 1}^{L}\; {W_{l} \cdot f_{l}^{2}}}},{\alpha_{3} = {\sum\limits_{l = 1}^{L}\; {W_{l} \cdot f_{l}^{3}}}},{\alpha_{4} = {\sum\limits_{l = 1}^{L}\; {W_{l} \cdot f_{l}^{4}}}},{\Phi_{1} = {\sum\limits_{l = 1}^{L}\; {{W_{l} \cdot \Delta}\; {I_{l} \cdot f_{l}}}}},{\Phi_{2} = {\sum\limits_{l = 1}^{L}\; {{W_{l} \cdot \Delta}\; {I_{l} \cdot f_{l}^{2}}}}},{h_{al} = \frac{W_{l} \cdot f_{l} \cdot \left( {\alpha_{4} - {\alpha_{3} \cdot f_{l}}} \right)}{{\alpha_{2}\alpha_{4}} - \alpha_{3}^{2}}},{h_{bl} = \frac{W_{l} \cdot f_{l} \cdot \left( {{\alpha_{2} \cdot f_{l}} - \alpha_{3}} \right)}{{\alpha_{2}\alpha_{4}} - \alpha_{3}^{2}}}}} & \left( {{Eq}.\mspace{14mu} 9} \right) \end{matrix}$

Note that:

$\begin{matrix} {{{\sum\limits_{l = 1}^{L}\; \left\lbrack {h_{al} \cdot f_{l}} \right\rbrack} = {\frac{{\alpha_{4} \cdot {\sum\limits_{l = 1}^{L}\; \left\lbrack {W_{l} \cdot f_{l}^{2}} \right\rbrack}} - {\alpha_{3} \cdot {\sum\limits_{l = 1}^{L}\; \left\lbrack {W_{l} \cdot f_{l}^{3}} \right\rbrack}}}{{\alpha_{2}\alpha_{4}} - \alpha_{3}^{2}} = {\frac{{\alpha_{4}\alpha_{2}} - \alpha_{3}^{2}}{{\alpha_{2}\alpha_{4}} - \alpha_{3}^{2}} = 1}}}{{\sum\limits_{l = 1}^{L}\; \left\lbrack {h_{al} \cdot f_{l}^{2}} \right\rbrack} = {\frac{{\alpha_{4} \cdot {\sum\limits_{l = 1}^{L}\; \left\lbrack {W_{l} \cdot f_{l}^{3}} \right\rbrack}} - {\alpha_{3} \cdot {\sum\limits_{l = 1}^{L}\; \left\lbrack {W_{l} \cdot f_{l}^{4}} \right\rbrack}}}{{\alpha_{2}\alpha_{4}} - \alpha_{3}^{2}} = {\frac{{\alpha_{4}\alpha_{3}} - {\alpha_{3}\alpha_{4}}}{{\alpha_{2}\alpha_{4}} - \alpha_{3}^{2}} = 0}}}{{\sum\limits_{l = 1}^{L}\; \left\lbrack {h_{bl} \cdot f_{l}} \right\rbrack} = {\frac{{\alpha_{2} \cdot {\sum\limits_{l = 1}^{L}\; \left\lbrack {W_{l} \cdot f_{l}^{3}} \right\rbrack}} - {\alpha_{3} \cdot {\sum\limits_{l = 1}^{L}\; \left\lbrack {W_{l} \cdot f_{l}^{2}} \right\rbrack}}}{{\alpha_{2}\alpha_{4}} - \alpha_{3}^{2}} = {\frac{{\alpha_{2}\alpha_{3}} - {\alpha_{3}\alpha_{2}}}{{\alpha_{2}\alpha_{4}} - \alpha_{3}^{2}} = 0}}}{{\sum\limits_{l = 1}^{L}\; \left\lbrack {h_{bl} \cdot f_{l}^{2}} \right\rbrack} = {\frac{{\alpha_{2} \cdot {\sum\limits_{l = 1}^{L}\; \left\lbrack {W_{l} \cdot f_{l}^{4}} \right\rbrack}} - {\alpha_{3} \cdot {\sum\limits_{l = 1}^{L}\; \left\lbrack {W_{l} \cdot f_{l}^{3}} \right\rbrack}}}{{\alpha_{2}\alpha_{4}} - \alpha_{3}^{2}} = {\frac{{\alpha_{2}\alpha_{4}} - \alpha_{3}^{2}}{{\alpha_{2}\alpha_{4}} - \alpha_{3}^{2}} = 1}}}} & \left( {{Eq}.\mspace{14mu} 10} \right) \end{matrix}$

As is made clear below, this property will be useful in the resist model section. The above set of equations can be readily generalized to accommodate a higher-order polynomial fitting.

A benefit of introducing the derivative images “a” and “b” is that using (Eq.4), the aerial image can be predicted at any point of the process window by straightforward scaling of the a and b images by the defocus offset and a simple addition, rather than performing a full image simulation (i.e., convolution of the mask pattern with the TCCs) at each particular defocus setting required for a PW analysis. In addition, changes in exposure dose can be expressed by a simple upscaling or downscaling of the image intensity by a factor (1−ε):

I(x,f,1+g)=(1+ε)·I(x,f)  (Eq.11)

where I(x,f) is the aerial image at the nominal exposure dose, while E is the relative change in dose.

Combining this with (Eq.4) yields the general result:

$\begin{matrix} \begin{matrix} {{I\left( {x,f,{1 + ɛ}} \right)} = {\left( {1 + ɛ} \right) \cdot \left\lbrack {{I_{0}(x)} + {{a(x)} \cdot \left( {f - f_{0}} \right)} + {{b(x)} \cdot \left( {f - f_{0}} \right)^{2}}} \right\rbrack}} \\ {= {{I_{0}(x)} + \left\lbrack {{ɛ \cdot {I_{0}(x)}} + {\left( {1 + ɛ} \right) \cdot {a(x)} \cdot \left( {f - f_{0}} \right)} + {\left( {1 + ɛ} \right) \cdot {b(x)} \cdot \left( {f - f_{0}} \right)^{2}}} \right\rbrack}} \\ {= {{I_{0}(x)} + {\Delta \; {I(x)}}}} \end{matrix} & \left( {{Eq}.\mspace{14mu} 12} \right) \end{matrix}$

where ΔI will typically be small perturbations within a reasonable range of PW parameter variations.

The foregoing method is characterized by a flow diagram in FIG. 3 where the contours, CD or Edge Placement Errors (EPEs) are to be extracted from the aerial image at different defocus conditions. Referring to FIG. 3, the first step (Step 40) in the process is to identify the target pattern or mask pattern to be simulated and the process conditions to be utilized. The next step (Step 42) is to generate a nominal image I_(O) and M defocus images {I_(l)} in accordance with (Eq.3) above. Thereafter, derivative images “a” and “b” are generated utilizing (Eq.9) (Step 43). The next step (Step 44) entails generating the defocus image utilizing (Eq.4), i.e., the synthesis of I₀, a (scaled by f) and b (scaled by f²). Next, contours are extracted and CDs or feature EPEs are determined from the simulated image (Step 46). The process then proceeds to Step 48 to determine whether or not there is sufficient coverage (e.g., whether it is possible to determine the boundary of the process window) and if the answer is no, the process returns to Step 44 and repeats the foregoing process. If there is sufficient coverage, the process is complete.

It is noted that if a sufficient coverage of the process window requires evaluation at N process window points, and L<N images are used for fitting the derivative images a and b, the reduction in computation time will be close to L/N, since scaling the predetermined images I₀, a and b requires significantly less computation time than an independent re-calculation of the projected image at each new parameter setting. The foregoing method is generally applicable, independent of the specific details of the aerial image simulation. Furthermore, it is also applicable to both the aerial image as well as to the resist image from which simulated resist contours are extracted.

The foregoing method also does not depend on any specific model or implementation used for simulating the set of aerial images {I₁, I₂, . . . , I_(L)} at varying defocus. However, the foregoing method requires a number L>2 of individual images to be simulated for each mask layout under consideration. In a second embodiment of the method of the present invention, an even more efficient solution is made possible by the TCC formalism introduced in (Eq.1).

From (Eq.1), each aerial image at focus f_(l)(l=0, 1, . . . , L) can be defined as:

I ₁(x)=Σ_(k′)Σ_(k″) TCC _(l,k′,k″) M(k′)M*(k″)exp(−j(k′−k″)x)

where TCC_(l) is the TCC at focus f_(l) and TCC_(l,k′,k″) is the matrix element of TCC_(l), and M() represents the mask image, which is independent of the focus.

Combining this with (Eq.9) and exchanging the order of summation,

$\begin{matrix} {\begin{matrix} {a = {\sum\limits_{l = 1}^{L}\; {h_{al}\left( {I_{l} - I_{0}} \right)}}} \\ {= {\sum\limits_{l = 1}^{L}\; {h_{al}\left( {{\Sigma_{k^{\prime}}\Sigma_{k^{''}}{TCC}_{l,k^{\prime},k^{''}}{M\left( k^{\prime} \right)}{M^{*}\left( k^{''} \right)}{\exp \left( {{- {j\left( {k^{\prime} - k^{''}} \right)}}x} \right)}} - {\Sigma_{k^{\prime}}\Sigma_{k^{''}}{TCC}_{0,k^{\prime},k^{''}}{M\left( k^{\prime} \right)}{M^{*}\left( k^{''} \right)}{\exp \left( {{- {j\left( {k^{\prime} - k^{''}} \right)}}x} \right)}}} \right.}}} \\ {= {\Sigma_{k^{\prime}}{\Sigma_{k^{''}}\left\lbrack {\sum\limits_{l = 1}^{L}\; {h_{al}\left( {{TCC}_{l,k^{\prime},k^{''}} - {TCC}_{0,k^{\prime},k^{''}}} \right)}} \right\rbrack}{M\left( k^{\prime} \right)}{M^{*}\left( k^{''} \right)}{\exp \left( {{- {j\left( {k^{\prime} - k^{''}} \right)}}x} \right)}}} \end{matrix}\begin{matrix} {b = {\sum\limits_{l = 1}^{L}\; {h_{bl}\left( {I_{l} - I_{0}} \right)}}} \\ {= {\sum\limits_{l = 1}^{L}\; {h_{bl}\left( {{\Sigma_{k^{\prime}}\Sigma_{k^{''}}{TCC}_{l,k^{\prime},k^{''}}{M\left( k^{\prime} \right)}{M^{*}\left( k^{''} \right)}{\exp \left( {{- {j\left( {k^{\prime} - k^{''}} \right)}}x} \right)}} - {\Sigma_{k^{\prime}}\Sigma_{k^{''}}{TCC}_{0,k^{\prime},k^{''}}{M\left( k^{\prime} \right)}{M^{*}\left( k^{''} \right)}{\exp \left( {{- {j\left( {k^{\prime} - k^{''}} \right)}}x} \right)}}} \right.}}} \\ {= {\Sigma_{k^{\prime}}{\Sigma_{k^{''}}\left\lbrack {\sum\limits_{l = 1}^{L}\; {h_{bl}\left( {{TCC}_{l,k^{\prime},k^{''}} - {TCC}_{0,k^{\prime},k^{''}}} \right)}} \right\rbrack}{M\left( k^{\prime} \right)}{M^{*}\left( k^{''} \right)}{\exp \left( {{- {j\left( {k^{\prime} - k^{''}} \right)}}x} \right)}}} \end{matrix}} & \left( {{Eq}.\mspace{14mu} 13} \right) \end{matrix}$

Thus if two new TCCs are defined as linear combinations of TCC_(l) (l=0, 1, . . . , L) in the following way:

$\begin{matrix} {{A = {{\sum\limits_{l = 1}^{L}\; {h_{al}\Delta \; {TCC}_{l}}} = {\sum\limits_{l = 1}^{L}\; {h_{al}\left( {{TCC}_{l} - {TCC}_{0}} \right)}}}},{B = {{\sum\limits_{l = 1}^{L}\; {h_{bl}\Delta \; {TCC}_{l}}} = {\sum\limits_{l = 1}^{L}\; {h_{bl}\left( {{TCC}_{l} - {TCC}_{0}} \right)}}}}} & \left( {{Eq}.\mspace{14mu} 14} \right) \end{matrix}$

then “a” and “b” are “aerial images” which can be computed directly from A and B, i.e.,

$\begin{matrix} {{{a(x)} = {\underset{k^{\prime}}{\Sigma}\underset{k^{''}}{\Sigma}A_{k^{\prime},k^{''}}{M\left( k^{\prime} \right)}{M^{*}\left( k^{''} \right)}{\exp \left( {{- {j\left( {k^{\prime} - k^{''}} \right)}}x} \right)}}}{{b(x)} = {\underset{k^{\prime}}{\Sigma}\underset{k^{''}}{\Sigma}B_{k^{\prime},k^{''}}{M\left( k^{\prime} \right)}{M^{*}\left( k^{''} \right)}{\exp \left( {{- {j\left( {k^{\prime} - k^{''}} \right)}}x} \right)}}}{{{where}\mspace{14mu} A_{k^{\prime},k^{''}}} = {{\sum\limits_{l = 1}^{L}\; {{h_{al}\left( {{TCC}_{l,k^{\prime},k^{''}} - {TCC}_{0,k^{\prime},k^{''}}} \right)}\mspace{14mu} {and}\mspace{14mu} B_{k^{\prime},k^{''}}}} = {\sum\limits_{l = 1}^{L}\; {h_{bl}\left( {{TCC}_{l,k^{\prime},k^{''}} - {TCC}_{0,k^{\prime},k^{''}}} \right)}}}}} & \left( {{Eq}.\mspace{14mu} 15} \right) \end{matrix}$

are the matrix elements of A and B, respectively. This also implies that a linear combination of aerial images of different planes can be computed using a single linear combination of TCCs corresponding to those planes.

A significant advantage of using TCC₀, A, and B in place of the L through-focus images is that the TCC₀, A, and B can be pre-computed, independently of the actual mask pattern, for known illumination and projection parameters, giving rise to the possibility of further reduction of computing time (down from L through-focus simulations for each mask pattern), which will be further explained below. It is noted that the generation of A and B neither requires computation of a set of aerial images at different defocus conditions nor requires calibration from this set of aerial images. Once TCC₀, A, and B have been calculated, these terms can be generally applied to predict the through-focus imaging performance for any specific mask design using (Eq.15) and (Eq.4). Besides the through-focus variation, a variation of exposure dose around nominal condition can be applied to the TCC terms by the same linear scaling as described by (Eq.11) and (Eq.12) above.

Calculating the derivative images a and b from TCCs A and B allows a further reduction of computation time by using only the dominant terms of A and B, as in the discussions related to (Eq.2). More specifically, suppose the diagonalization of TCC₀, A and B is:

$\begin{matrix} {{{TCC}_{0} = {\sum\limits_{i = 1}^{N_{0}}\; {\lambda_{0,i}{\varphi_{0,i}\left( k^{\prime} \right)}{\varphi_{0,i}\left( k^{''} \right)}}}}{A = {\sum\limits_{i = 1}^{N_{A}}\; {\lambda_{A,i}{\varphi_{A,i}\left( k^{\prime} \right)}{\varphi_{A,i}\left( k^{''} \right)}}}}{B = {\sum\limits_{i = 1}^{N_{B}}\; {\lambda_{B,i}{\varphi_{B,i}\left( k^{\prime} \right)}{\varphi_{B,i}\left( k^{''} \right)}}}}} & \left( {{Eq}.\mspace{14mu} 16} \right) \end{matrix}$

where λ_(0,i)(i=1, . . . , N₀) denotes the N₀ largest eigenvalues and φ_(0,i)() denotes the corresponding eigenvector of the TCC matrix TCC₀; λ_(A,i)(i=1, . . . , N_(A)) denotes the N_(A) largest eigenvalues and φ_(A,i)() denotes the corresponding eigenvector of the TCC matrix A; and λ_(B,i)(i=1, . . . , N_(B)) denotes the N_(B) largest eigenvalues and φ_(B,i)() denotes the corresponding eigenvector of the TCC matrix B. Then, from (Eq.3), for mask image M(),

$\begin{matrix} {{I_{0}(x)} = {\left. {\sum\limits_{i = 1}^{N_{0}}\; \lambda_{0,i}} \middle| {\Phi_{0,i}(x)} \middle| {}_{2}{a(x)} \right. = {\left. {\sum\limits_{i = 1}^{N_{A}}\; \lambda_{A,i}} \middle| {\Phi_{A,i}(x)} \middle| {}_{2}{b(x)} \right. = \left. {\sum\limits_{i = 1}^{N_{B}}\; \lambda_{B,i}} \middle| {\Phi_{B,i}(x)} \right|^{2}}}} & \left( {{Eq}.\mspace{14mu} 17} \right) \end{matrix}$

where I₀ is the nominal aerial image,

${{\Phi_{0,i}(x)} = {\underset{k^{''}}{\Sigma}{\varphi_{0,i}\left( k^{''} \right)}{M\left( k^{''} \right)}{\exp \left( {{- {jk}^{''}}x} \right)}}},{{\Phi_{A,i}(x)} = {{\underset{k^{''}}{\Sigma}{\varphi_{A,i}\left( k^{''} \right)}{M\left( k^{''} \right)}{\exp \left( {{- {jk}^{''}}x} \right)}\mspace{14mu} {and}\mspace{14mu} {\Phi_{B,i}(x)}} = {\underset{k^{''}}{\Sigma}{\varphi_{B,i}\left( k^{''} \right)}{M\left( k^{''} \right)}{{\exp \left( {{- {jk}^{''}}x} \right)}.}}}}$

Utilizing a larger number of TCC terms can improve the accuracy of the optical model and the separability of optical and resist model components. However, since the image or TCC derivatives relate to relatively minor image variations within the PW, typically on the order of 10% in CD variation, a smaller number of terms may suffice for the A and B terms than for the Nominal Condition TCC₀. For example, if 64 terms are considered for TCC₀, (i.e., N₀=64), only 32 terms are typically required for each of the A and B terms in order to achieve sufficient CD prediction accuracy, i.e., N_(A)=N_(B)=32. In this case, approximately the same amount of computation time will be required to generate the derivative images a and b as compared to the nominal condition I₀. It is noted that, unlike the original TCC matrices, a coefficient TCC matrix such as A or B is in general not non-negative-definite, which implies both positive and negative eigenvalues exist for a derivative TCC matrix. Therefore, the leading terms from the eigen-series expansion and truncation should include all eigenvalues with the largest absolute values, both positive and negative.

Similar to (Eq.5), (Eq.14) can be derived alternatively from series expansion. More specifically, the variation of TCC matrix elements around nominal or best focus f₀ may also be expressed as a series expansion:

$\begin{matrix} {{{TCC}_{k^{\prime},k^{''}}(f)} = \left. {{{TCC}_{k^{\prime},k^{''}}\left( f_{0} \right)} + \frac{\partial{{TCC}_{k^{\prime},k^{''}}(f)}}{\partial f}} \middle| {}_{f = f_{0}}{{\cdot \left( {f - f_{0}} \right)} + {2\frac{\partial^{2}{{TCC}_{k^{\prime},k^{''}}(f)}}{\partial f^{2}}}} \middle| {}_{f = f_{0}}{\cdot \left( {f -_{0}} \right)^{2}} \right.} & \left( {{Eq}.\mspace{14mu} 18} \right) \end{matrix}$

Thus, the coefficients of the series expansion may be evaluated directly by a numerical finite difference method, or again from a least-square fitting to a number of individually calculated TCC terms corresponding to a set of focus positions, in a manner similar to the through-focus fitting of aerial images discussed in the previous section. The fitting approach provides a wider range of validity, and introduces weight factors to place more or less emphasis on certain parts of the PW. This approach will follow (Eq.6)-(Eq.9) after replacing the set of test images I_(l) by their corresponding TCCs in the equations. Consequently, the best fit derivative matrices A and B are obtained from the same linear combination set forth above, also after formally replacing the I_(l) by TCC_(l), i.e.,

$\begin{matrix} {{A = {{\sum\limits_{l = 1}^{L}\; {h_{al}\Delta \; {TCC}_{l}}} = {\sum\limits_{l = 1}^{L}\; {h_{al}\left( {{TCC}_{l} - {TCC}_{0}} \right)}}}},{B = {{\sum\limits_{l = 1}^{L}\; {h_{bl}\Delta \; {TCC}_{l}}} = {\sum\limits_{l = 1}^{L}\; {h_{bl}\left( {{TCC}_{l} - {TCC}_{0}} \right)}}}}} & \left( {{Eq}.\mspace{14mu} 19} \right) \end{matrix}$

where h_(al) and h_(bl) are again computed using (Eq.9). It is noted that h_(al) and h_(bl) are constants that do not depend on the patterns or TCC_(l). Thus, A and B are simply a linear combination of the Nominal Condition TCC₀ and a set of TCCs at various defocus conditions (TCC₁ through TCC_(L)).

It will be appreciated that since (Eq.19) is the same as (Eq.14), the two alternative approaches lead to the same final formulation. Similarly, (Eq.4) can also be derived from (Eq.15), (Eq.18), and (Eq.19).

Certain embodiments employ a method illustrated by the example provided in the flow diagram of FIG. 4 where the contours, CD or Edge Placement Errors (EPEs) are to be extracted from the aerial image at different defocus conditions. The first step (Step 50) in the process is to identify the process specific optical conditions associated with the desired process. The next step (Step 52) is to generate a nominal condition TCC₀ and L defocus {TCC_(l)}. Thereafter, derivative TCCs: A and B are generated utilizing (Eq.14) (Step 54). The next step (Step 58) generates images I₀, a, b by convolution of the mask image with TCC₀, A and B utilizing (Eq.17). Next, for each mask design (Step 56), defocus image is synthesized utilizing (Eq.4) (Step 60), thereby generating the simulated image. Next, contours are extracted and CDs or feature EPEs are determined from the simulated image (Step 62). The process then proceeds to Step 64 to determine whether or not there is sufficient coverage to determine the boundary of the process window and if the answer is no, the process returns to Step 58 and repeats the foregoing process. If there is sufficient coverage, the process proceeds to Step 66 to determine if the image produced by the mask design is within allowable error tolerances, and if so, the process is complete. If not, the process returns to Step 56 so as to allow for adjustment and redesign of the mask. It is noted that this last step is an optional step in the process.

The flowchart depicted in FIG. 4 shows an example of PW analysis embedded within a “mask variation loop” which may be required for iterative, PW-aware OPC modifications of an initial mask design. In this situation, any improvement in computation speed for the through-PW image assessment will be especially beneficial.

An additional reduction in computation time may be achieved by further suitable assumptions or a priori knowledge about the physics of the optical system. For example, in the absence of strong aberrations, it can be expected that the through-focus variation of aerial image intensities will be an even (i.e. symmetrical) function of defocus. Therefore, it can be expected that the first-order derivatives “A” and “a” will be negligible under these conditions.

This simplification can be further justified by noting that the effect of defocus corresponds to a multiplication of the pupil function by a phase factor p=p₀ exp [ja(f−f₀)²], where the nominal focus is at f₀=0. For small defocus the phase shift can be approximated by a Taylor expansion: p=p₀. [1+ja(f−f₀)²], which does not contain a linear term.

All the above methods may also be extended to a generalized process window definition that can be established by different or additional base parameters in addition to exposure dose and defocus. These may include optical settings such as NA, sigma, aberrations, polarization, or optical constants of the resist layer (whose effects on the imaging process are included in the optical model, i.e. the TCCs). In one example, including a variation of NA around nominal conditions, the aerial image can be expressed as:

I(f,NA)=I ₀ +a·(f−f ₀)+b·(f−f ₀)² +c·(NA−NA ₀)+d·(NA−NA ₀)² +e·(f−f ₀)·(NA−NA ₀)  (Eq.20)

where I, I₀, a, . . . , e are 2-dimensional images and image derivatives, respectively. The additional parameters “c”, “d”, and “e” can be determined by a least square fit to a set of simulated images or a set of simulated TCCs at varying parameter values for f and NA, while the scaling with exposure dose as in (Eq.11) and (Eq.12) still applies. Similar to (Eq.9), these parameters (a, b, c, d, and the cross-term coefficient e) are again a linear combination of aerial images {I_(l)}. The coefficients of this linear combination do not depend on the pixel coordinate or pattern, but only on the values of the {f_(l)}, {NA_(l)}, and/or the user-assigned weights {W_(l)}.

For this generalized PW model, simplifications based on physical insight are also possible. In case of NA variations, for example, it can be expected that these will have a rather monotonous, linear effect on the image variations, in which case (Eq.20) can be simplified by dropping the higher order “d” and “e” terms in NA, possibly in addition to the linear term in defocus. Also, for any generalized PW definition, the number of TCC terms used for calculating I₀ at Nominal Condition need not be the same as the number of terms used for calculating image variations from the TCC derivatives A, B, . . . A sufficiently accurate description of minor image variations due to small parameter variations around Nominal Condition may be achieved with a large number of terms for I₀ and a significantly smaller number for the derivatives, in order to reduce the overall computation time.

For simplicity purposes, the following discussion will be based on defocus and exposure dose. However, it should be noted that all the disclosures herein can be extended to generalized PW with other parameters such as NA, sigma, aberrations, polarization, or optical constants of the resist layer, as illustrated in (Eq.20). In the examples described above, analytic expressions for the aerial image in the vicinity of best focus for a range of PW parameters were developed. The following descriptions derive similar expressions and methods to calculate the resist image, which forms the basis for extraction of simulated resist contours across the PW.

Separable, Linear Resist Model

Although the response of photo resist to illumination by the projected aerial image may be strongly nonlinear, having a thresholding behavior, many processes occurring in the resist layer, such as diffusion during PEB, can be modeled by convoluting the aerial image with one or more linear filters before applying the threshold. Such models will be generally referred to as “linear” resist models, and the latent resist image for such models may be expressed schematically as:

R(x)=P{I(x)}+R _(b)(x)  (Eq.21)

Here, P{ } denotes the functional action of applying a linear filter (i.e. generally a convolution), while Rb is a mask loading bias that is independent of the aerial image. The resist threshold is understood to be included in R_(b) such that resist contours correspond to locations where R(x)=0.

Applying this model to the general, scaled, interpolated aerial image derived above, i.e., (Eq.12, assuming f₀=0 without loss of generality), results in

$\begin{matrix} \begin{matrix} {R = {\left\lbrack {{P\left\{ I_{0} \right\}} + R_{b}} \right\rbrack + {{ɛ \cdot P}\left\{ I_{0} \right\}} + {{\left( {1 + ɛ} \right) \cdot f \cdot P}\left\{ a \right\}} + {{\left( {1 + ɛ} \right) \cdot f^{2} \cdot P}\left\{ b \right\}}}} \\ {= {R_{0} + {{ɛ \cdot P}\left\{ I_{0} \right\}} + {{\left( {1 + ɛ} \right) \cdot f \cdot P}\left\{ a \right\}} + {{\left( {1 + ɛ} \right) \cdot f^{2} \cdot P}\left\{ b \right\}}}} \\ {= {R_{0} + {\Delta \; {R\left( {x,ɛ,f} \right)}}}} \end{matrix} & \left( {{Eq}.\mspace{14mu} 22} \right) \end{matrix}$

where R₀ is the resist image at Nominal Condition (NC). All corrections due to changes in exposure dose and focus (or, other PW parameters) can be derived by applying the same filter to the derivative images a, b as to the image I₀ at NC, and simple scaling and summation of the correction terms.

Moreover, the effect of a linear filter may be included in the imaging TCC formalism, since the convolution with a filter in the space domain is equivalent to a multiplication with the filter's Fourier series components in the frequency domain. Starting from an aerial image expression (Eq.1):

I(x)=Σ_(k′)Σ_(k″) TCC _(k′,k″) M(k′)M*(k″)exp(−j(k′−k″)x)

It is shown that the TCC matrix element at k, k″contributes to the (k′-k″) frequency component of I(x) by the amount TCC_(k′, k″)M(k′)M*(k″). Therefore, a resist image defined by:

I(x)

g(x)

where g(x) is a spatial filter with the Fourier transform being G(k), can be expressed as:

$\begin{matrix} {{{I(x)} \otimes {g(x)}} = {\Sigma_{k^{\prime}}\Sigma_{k^{''}}{TCC}_{k^{\prime},k^{''}}{M\left( k^{\prime} \right)}{M^{*}\left( k^{''} \right)}{\exp \left( {{- {j\left( {k^{\prime} - k^{''}} \right)}}x} \right)}{G\left( {k^{\prime} - k^{''}} \right)}}} \\ {= {\Sigma_{k^{\prime}}\Sigma_{k^{''}}{TCC}_{k^{\prime},k^{''}}^{new}{M\left( k^{\prime} \right)}{M^{*}\left( k^{''} \right)}{\exp \left( {{- {j\left( {k^{\prime} - k^{''}} \right)}}x} \right)}}} \end{matrix}$

with a new TCC matrix defined as TCC^(new) _(k′,k″)=TCC_(k′,k″)G(k′−k″).

With this procedure, the linear filter is incorporated into the bi-linear TCC matrix, so all the computational procedures applicable to a purely optical aerial image may be applied to the linearly filtered aerial image. This property allows a significant reduction in overall computation time, since the complete resist image can be generated by a single evaluation of (Eq.1), with the only modification of adding weight factors corresponding to the Fourier coefficients of the filter P. For any given mask design input, this formulation would allow to generate directly, in one pass each, the images P{I₀}, P{a}, P{b} from the pre-computed, filter-adjusted TCC₀, A, and B matrices. (Eq.22) then defines the actual resist image for any arbitrary PW point as a linear combination of these three images.

Non-Separable, Linear Resist Model

In the preceding discussion, it was implicitly assumed that all parameters of the linear filters establishing the resist model are constant across the variations of the process window parameters. This equates to one condition for an overall separable lithography model: resist model parameters are independent of optical model parameters. A pragmatic test for separability is the ability to accurately calibrate the model and fit test data across the complete extent of the PW. In practice, the semi-empirical nature of models suitable for full-chip lithography simulation may preclude perfect separability and may require resist model parameters that are allowed to vary with PW parameters such as defocus, NA or sigma settings. For a physically motivated model, it should be expected (or required as a constraint), though that the model parameters vary smoothly under variation of the PW variables. In this case, the series expansion of the resist image may include derivative terms of the resist model parameters.

For illustration purposes, consider defocus as the only PW parameter. If the linear resist model is equivalent to a convolution with a linear filter, (or a multitude of linear filters), a separable model may be described by:

R(x,f)=P(x)

I(x,f)+R _(b)(x)  (Eq.23)

while a non-separable model may require an explicit f-dependence of the filter

R(x,f)=P(x,f)

I(x,f)+R _(b)(x)  (Eq.24)

Now, considering through-focus changes, a pro-forma series expansion may be applied to (Eq.24), for illustration herein only up to first order:

$\begin{matrix} {\begin{matrix} {{R\left( {x,f} \right)} = {{R\left( {x,f_{0}} \right)} + {\left\lbrack {{{a_{P}(x)} \otimes {I_{0}(x)}} + {{P\left( {x,f_{0}} \right)} \otimes {a(x)}}} \right\rbrack \cdot \left( {f - f_{0}} \right)} + \ldots}} \\ {= {{R_{0}(x)} + {\Delta \; {R\left( {x,f} \right)}}}} \end{matrix}{where}{{a_{P}(x)} = \left. \frac{\partial{P\left( {x,f} \right)}}{\partial f} \right|_{f = f_{0}}}} & \left( {{Eq}.\mspace{14mu} 25} \right) \end{matrix}$

(Eq.26)

If the resist model parameters are found to vary continuously across the PW space, similar series expansion and fitting as introduced above for the AI and TCCs can be applied to the resist model parameters during model calibration. In this case a linear, derivative filter a_(P) can be calculated and be used in (Eq.25), which may also be extended in a straightforward way to include higher-order terms. In this situation, resist model parameters as well as aerial image variations are smoothly interpolated across the complete PW area. Both P and a_(P) can be determined in a through-PW model calibration step based on experimental wafer data from test or gauge patterns.

However, even if resist model parameters appear to vary non-monotonously across the PW, any piece-wise interpolation in between calibration points could provide ‘best-guess’ resist model parameters for arbitrary PW points.

General Resist Model

For a general resist model that may include nonlinear operations such as truncations of the aerial or resist image, the straightforward separation into nominal condition and derivative terms, as shown in (Eq.22) will be no longer valid. However, there are three alternative methods to deal with the non-linear operations.

i) Associated Linear Filter

First, it is assumed that the general variation of the resist image through PW can be approximated formally by the second line in (Eq.22), with the reinterpretation that the linear filter P{ } will no longer correctly describe the resist model at NC (Normal Condition). Instead, linear filter P{ } will be chosen to reproduce the best representation of differential resist image changes relative to the NC. While a nonlinear model may ensure the most accurate model fitting at the NC, it may require significantly more computation time than a linear model. By relying on such an associated linear filter to emulate the differential through-PW behavior, only a single evaluation of the nonlinear model will be required to generate R₀(x), while PW analysis at a multitude of PW conditions can be based on more efficient evaluation of P{I₀}, P{a}, P{b}.

The coefficients of the nominal condition resist model as well as of the associated filter may be determined from a unified model calibration procedure based on calibration test patterns and wafer gauge data covering pattern variations and process window variations, as an extension of the method described in U.S. Patent Application No. 60/719,837.

Further, once a valid unified PW model (FEM) has been generated and calibrated in the manner set forth in U.S. Patent Application No. 60/719,837, it will provide the best prediction of through-PW changes of the resist image. The parameters of the optimum associated filter may then be determined by minimizing the overall (RMS (root mean square)) difference between the simplified model using the associated filter and the complete, calibrated model, without any need for additional experimental calibration data.

Using the full model, for any suitable number and range of test structures, including e.g. 1-D (line/space) and 2-D (line ends etc) patterns, ‘correct’ resist images and contours can be simulated for any number of PW points. In addition, the values of the derivative images a and b can be calculated in the vicinity of the resist contours. For each pattern, the change of R(x) through-PW will be calculated at pattern-specific gauge points, e.g. the tip of a line for a line-end test pattern, or along any point of the NC resist contour. At each of these evaluation points x_(i) through

ΔR(x _(i) ,ε,f)=R(x _(i) ,ε,f)−R(x _(i),ε=0,f=f ₀)=R(x _(i) ,ε,f),  (Eq.27)

since x_(i) is assumed to be on a resist contour, where R(x_(i),ε=0,f=f₀)=0. ΔR(x_(i),ε,f) should be well approximated by:

ΔR _(a)(x _(i))=ε·P{I ₀(x _(i))}+(1+ε)·f·P{a(x _(i))}+(1+ε)·f ² ·P{b(x _(i))}.  (Eq.28)

Therefore, the optimal associated filter will minimize the sum of squared differences between (Eq.27) and (Eq.28) and can be determined by various known optimization algorithms. It is noted that evaluation of (Eq.27) and (Eq.28) during the associated filter fitting should be performed at resist contours, so that the resulting filter most closely reproduces changes close to edge positions. Performance of the associated filter—in terms of accurately predicting changes in the resist image level—far away from edge positions is generally not required. After this fitting routine, the full-PW behavior of the resist images is again described as

R(x,ε,f)=R ₀(x)+ΔR _(a)(x,ε,f)  (Eq.29)

where the filtered differential images can be efficiently calculated within the TCC formalism, the ΔR constitutes relatively small perturbations, and the resist images at any arbitrary PW point can be predicted from a simple linear combination of the four images R₀, P{I₀}, P{a}, and P{b}.

ii) Embedded Linearization

The above approach presents a linearized filter (i.e., the associated filter) which is optimal in that it is the single linear filter which minimizes the (RMS) difference for all pattern-specific gauge points or along any point of the NC (Nominal Condition) resist contour. Next, an alternative approach is discussed which incorporates resist model linearization in the computation of derivative resist images.

More specifically, after obtaining a and b in (Eq.2), the goal becomes identifying R₀, Ra and Rb such that their linear combination (assuming that f₀=0 without loss of generality)

R _(EL)(x,f)=R ₀(x)+Ra(x)·f+Rb(x)·f ²  (Eq.30)

is the best fit for

$\begin{matrix} \begin{matrix} {{R\left( {x,f_{l}} \right)} = {R\left\{ {I\left( {x,f_{l}} \right)} \right\}}} \\ {= {R\left\{ {{I_{0}(x)} + {{a(x)} \cdot f_{l}} + {{b(x)} \cdot f_{l}^{2}}} \right\}}} \end{matrix} & \left( {{Eq}.\mspace{14mu} 31} \right) \end{matrix}$

over a number of focus positions f_(l)={f₁, f₂, . . . , f_(L)} with possibly a set of weights {W₁, W₂, . . . , W_(L)}, where R₀ is the resist image at NC. (Eq.31) is essentially applying the resist model R{•} to the aerial image expressed in (Eq.2). The resist model R{•} may be non-linear, thus Ra and Rb are not necessarily P{a} and P{b} or R{a} and R{b}. As such:

$\begin{matrix} {{{R_{0}(x)} = {R\left( {I_{0}(x)} \right)}}{{{Ra}(x)} = {\sum\limits_{l = 1}^{L}\; {h_{al}\left\lbrack {{R\left( {x,f_{l}} \right)} - {R_{0}(x)}} \right\rbrack}}}{{{Rb}(x)} = {\sum\limits_{l = 1}^{L}\; {h_{bl}\left\lbrack {{R\left( {x,f_{l}} \right)} - {R_{0}(x)}} \right\rbrack}}}} & \left( {{Eq}.\mspace{14mu} 32} \right) \end{matrix}$

where h_(al) and h_(bl) are coefficients defined in (Eq.9). The coefficients only depend on {f₁, f₂, . . . , f_(L)} and possibly weights {W₁, W₂, . . . , W_(L)}, and they are independent of R(x, f_(l)) or I(x, f_(l)).

In general, the resist model R{•} can be separated as:

R{I(x)}=P{I(x)}+P _(NL) {I(x)}+R _(b)  (Eq.33)

where Rb is a mask loading bias that is independent of the aerial image I(x) or focus, P{ } is the linear filter operation and P_(NL){ } is some non-linear operation. Combining (Eq.32) and (Eq.33),

$\begin{matrix} {\begin{matrix} {{{Ra}(x)} = {\sum\limits_{l = 1}^{L}\; {h_{al}\left\lbrack {{R\left( {x,f_{l}} \right)} - {R_{0}(x)}} \right\rbrack}}} \\ {= {{\sum\limits_{l = 1}^{L}\; {h_{al}\left\lbrack {{P\left\{ {I\left( {x,f_{l}} \right)} \right\}} - {P\left\{ {I_{0}(x)} \right\}}} \right\rbrack}} + {\sum\limits_{l = 1}^{L}\; {h_{al}\left\lbrack {{P_{NL}\left\{ {I\left( {x,f_{l}} \right)} \right\}} - {P_{NL}\left\{ {I_{0}(x)} \right\}}} \right\rbrack}}}} \end{matrix}\begin{matrix} {{{Rb}(x)} = {\sum\limits_{l = 1}^{L}\; {h_{bl}\left\lbrack {{R\left( {x,f_{l}} \right)} - {R_{0}(x)}} \right\rbrack}}} \\ {= {{\sum\limits_{l = 1}^{L}\; {h_{bl}\left\lbrack {{P\left\{ {I\left( {x,f_{l}} \right)} \right\}} - {P\left\{ {I_{0}(x)} \right\}}} \right\rbrack}} + {\sum\limits_{l = 1}^{L}\; {h_{bl}\left\lbrack {{P_{NL}\left\{ {I\left( {x,f_{l}} \right)} \right\}} - {P_{NL}\left\{ {I_{0}(x)} \right\}}} \right\rbrack}}}} \end{matrix}} & \left( {{Eq}.\mspace{14mu} 34} \right) \end{matrix}$

As discussed previously, since P{ } is a linear operation, then

$\begin{matrix} \begin{matrix} {{P\left\{ {I\left( {x,f_{l}} \right)} \right\}} = {P\left\{ {{I_{o}(x)} + {{a(x)} \cdot f_{l}} + {{b(x)} \cdot f_{l}^{2}}} \right\}}} \\ {= {{P\left\{ {I_{0}(x)} \right\}} + {P{\left\{ {a(x)} \right\} \cdot f_{l}}} + {P{\left\{ {b(x)} \right\} \cdot f_{l}^{2}}}}} \end{matrix} & \left( {{Eq}.\mspace{14mu} 35} \right) \end{matrix}$

As expected, it is possible to derive the following result with the aid of (Eq.9) and (Eq.10) set forth above,

$\begin{matrix} \begin{matrix} {{\sum\limits_{l = 1}^{L}\; {h_{al}\left\lbrack {{P\left\{ {I\left( {x,f_{l}} \right)} \right\}} - {P\left\{ {I_{0}(x)} \right\}}} \right\rbrack}} =} & {{\sum\limits_{l = 1}^{L}\; {h_{al}\left\lbrack {{P{\left\{ {a(x)} \right\} \cdot f_{l}}} + {P{\left\{ {b(x)} \right\} \cdot f_{l}^{2}}}} \right\rbrack}}} \\ {=} & {{{P{\left\{ {a(x)} \right\} \cdot {\sum\limits_{l = 1}^{L}\; \left\lbrack {h_{al} \cdot f_{l}} \right\rbrack}}} + {P{\left\{ {b(x)} \right\} \cdot {\sum\limits_{l = 1}^{L}\; \left\lbrack {h_{al} \cdot f_{l}^{2}} \right\rbrack}}}}} \\ {=} & {{P\left\{ {a(x)} \right\}}} \\  & {{\sum\limits_{l = 1}^{L}\; {h_{bl}\left\lbrack {{P\left\{ {I\left( {x,f_{l}} \right)} \right\}} - {P\left\{ {I_{0}(x)} \right\}}} \right\rbrack}}} \\ {=} & {{\sum\limits_{l = 1}^{L}\; {h_{bl}\left\lbrack {{P{\left\{ {a(x)} \right\} \cdot f_{l}}} + {P{\left\{ {b(x)} \right\} \cdot f_{l}^{2}}}} \right\rbrack}}} \\ {=} & {{{P{\left\{ {a(x)} \right\} \cdot {\sum\limits_{l = 1}^{L}\; \left\lbrack {h_{bl} \cdot f_{l}} \right\rbrack}}} + {P{\left\{ {b(x)} \right\} \cdot {\sum\limits_{l = 1}^{L}\; \left\lbrack {h_{bl} \cdot f_{l}^{2}} \right\rbrack}}}}} \\ {=} & {{P\left\{ {b(x)} \right\}}} \end{matrix} & \left( {{Eq}.\mspace{14mu} 36} \right) \end{matrix}$

Thus, Ra and Rb can computed from

$\begin{matrix} {{{{Ra}(x)} = {{P\left\{ {a(x)} \right\}} + {\sum\limits_{l = 1}^{L}\; {h_{al}\left\lbrack {{P_{NL}\left\{ {I\left( {x,f_{l}} \right)} \right\}} - {P_{NL}\left\{ {I_{0}(x)} \right\}}} \right\rbrack}}}}{{{Rb}(x)} = {{P\left\{ {b(x)} \right\}} + {\sum\limits_{l = 1}^{L}\; {h_{bl}\left\lbrack {{P_{NL}\left\{ {I\left( {x,f_{l}} \right)} \right\}} - {P_{NL}\left\{ {I_{0}(x)} \right\}}} \right\rbrack}}}}} & \left( {{Eq}.\mspace{14mu} 37} \right) \end{matrix}$

The benefits of this approach are that it does not attempt to capture the differential through-PW behavior for all gauge points using a single linear filter. Rather, this approach minimizes the (RMS) difference for each pixel, thereby improving the overall accuracy. In addition, this approach does not require identification of pattern-specific gauge points or all NC resist contour neighboring points. One drawback is that this approach slightly increases the computation complexity for Ra and Rb. However, since the synthesis of through-PW resist images only require scaling and additions of R₀, Ra and Rb, the increase in the computation complexity of the derivative images is generally insignificant compared to the reduction in computation complexity of through-PW resist images, especially for dense PW sampling.

iii) Polynomial Approximation of Non-Linear Operations

In a third approach, non-linear resist model operations are approximated using polynomials. More specifically, for truncation operations on image I(x), for the purpose of emulating acid and base reaction effects, quadratic polynomials of the image provide a sufficient approximation. Another typical non-linear operation, the linear filtering of the image slope, can be expressed precisely as the linear filtering of a quadratic function of the image gradient G{I(x)}=I(x)−I(x−1), thus the quadratic polynomial of the aerial image I(x) itself. More specifically, letting G{ } be the gradient operation and the linear filter be P_(Slope){}, then this non-linear operation can be expressed as:

P _(Slope) {G{I(x)}}=_(Slope){(I(x)−I(x−1))²}  (Eq.38)

To summarize, the resist image from aerial image I(x) can be approximated as:

$\begin{matrix} \begin{matrix} {{R\left\{ {I\left( {x,f} \right)} \right\}} =} & {{{P_{1}\left\{ {I\left( {x,f} \right)} \right\}} + {P_{2}\left\{ {I^{2}\left( {x,f} \right)} \right\}} + {R_{b}(x)} + {P_{Slope}\left\{ \left( {{I\left( {x,f} \right)} - {I\left( {{x - 1},f} \right)}} \right)^{2} \right\}}}} \\ {=} & {{{P_{1}\left\{ {{I_{0}(x)} + {{a(x)} \cdot f} + {{b(x)} \cdot f^{2}}} \right\}} + {P_{2}\left\{ \left( {{I_{0}(x)} + {{a(x)} \cdot f} + {{b(x)} \cdot f^{2}}} \right)^{2} \right\}} + {R_{b}(x)} +}} \\  & {{P_{Slope}\left\{ \left( {{I_{0}(x)} + {{a(x)} \cdot f} + {{b(x)} \cdot f^{2}} - {I_{0}\left( {x - 1} \right)} - {{a\left( {x - 1} \right)} \cdot f} - {{b\left( {x - 1} \right)} \cdot f^{2}}} \right)^{2} \right\}}} \\ {=} & {{{P_{1}\left\{ {I_{0}(x)} \right\}} + {P_{1}{\left\{ {a(x)} \right\} \cdot f}} + {P_{1}{\left\{ {b(x)} \right\} \cdot f^{2}}} + {P_{2}\left\{ {I_{0}^{2}(x)} \right\}}}} \\  & {{{{+ 2}P_{2}{\left\{ {{a(x)} \cdot {I_{0}(x)}} \right\} \cdot f}} + {P_{2}{\left\{ {{2{{b(x)} \cdot {I_{0}(x)}}} + {a^{2}(x)}} \right\} \cdot f^{2}}}}} \\  & {{{{+ 2}P_{2}{\left\{ {{a(x)} \cdot {b(x)}} \right\} \cdot f^{3}}} + {P_{2}{\left\{ {b^{2}(x)} \right\} \cdot f^{4}}} + {R_{b}(x)}}} \\  & {{{+ P_{Slope}}\left\{ \left( {{G\left\{ I_{0} \right\} (x)} + {G\left\{ a \right\} {(x) \cdot f}} + {G\left\{ b \right\} {(x) \cdot f^{2}}}} \right)^{2} \right\}}} \\ {=} & {\left\{ {{P_{1}\left\{ {I_{0}(x)} \right\}} + {P_{2}\left\{ {I_{0}^{2}(x)} \right\}} + {P_{Slope}\left\{ {G^{2}\left\{ I_{0} \right\} (x)} \right\}} + {R_{b}(x)}} \right\}} \\  & {{{+ \left\{ {{P_{1}\left\{ {a(x)} \right\}} + {2P_{2}\left\{ {{a(x)} \cdot {I_{0}(x)}} \right\}} + {2P_{Slope}\left\{ {G\left\{ a \right\} {(x) \cdot G}\left\{ I_{0} \right\} (x)} \right\}}} \right\}} \cdot f}} \\  & {{{+ \left\{ {{P_{1}\left\{ {b(x)} \right\}} + {P_{2}\left\{ {{2{{b(x)} \cdot {I_{0}(x)}}} + {a^{2}(x)}} \right\}} + {P_{Slope}\left\{ {{2G\left\{ a \right\} {(x) \cdot G}\left\{ I_{0} \right\} (x)} + {G^{2}\left\{ a \right\} (x)}} \right\}}} \right\}} \cdot f^{2}}} \\  & {{{+ 2}{\left\{ {{P_{2}\left\{ {{a(x)} \cdot {b(x)}} \right\}} + {P_{Slope}\left\{ {G\left\{ a \right\} {(x) \cdot G}\left\{ b \right\} (x)} \right\}}} \right\} \cdot f^{3}}}} \\  & {{{+ \left\{ {{P_{2}\left\{ {b^{2}(x)} \right\}} + {P_{Slope}\left\{ {G^{2}\left\{ b \right\} (x)} \right\}}} \right\}} \cdot f^{4}}} \\ {=} & {{{R_{0}(x)} + {{R_{1}(x)} \cdot f} + {{R_{2}(x)} \cdot f^{2}} + {{R_{3}(x)} \cdot f^{3}} + {{R_{4}(x)} \cdot f^{4}}}} \end{matrix} & \left( {{Eq}.\mspace{14mu} 39} \right) \end{matrix}$

Once again, P₁{} represents the linear filter for the aerial image term, P₂{} represents the linear filter for the aerial image square term, and P_(Slope){} represents the linear filter for the aerial image gradient term, while Rb is a mask loading bias that is independent of the image pattern. Thus the resist image is expressed as a 4^(th)-order polynomial of the defocus value. However, in a typical application, R₃(x) and R₄(x) are very small and can be ignored to improve the computational efficiency.

As noted above, the goal of lithography design verification is to ensure that printed resist edges and line widths are within a pre-specified distance from the design target. Similarly, the size of the process window—exposure latitude and depth of focus—are defined by CDs or edge placements falling within the specified margin. The various methods outlined above provide very efficient ways to determine the change of resist image signal level with variation of focus and exposure dose or other, generalized PW parameters. Each method resulted in an approximate expression of through-PW resist image variations ΔR as perturbation of the NC (Nominal Condition) image R₀.

In order to relate these changes in R(x) to changes in edge placement, in most cases a first-order approximation will suffice, due to the small CD or edge placement tolerances. Therefore, the lateral shift of any resist contour (R=0) (i.e., the edge placement change) is simply approximated by the image gradient G at the original (i.e. NC) contour location and the change in resist image level ΔR due to variation of focus, dose, etc. as:

$\begin{matrix} {{\Delta \; {{EP}\left( {x_{i},ɛ,f} \right)}} = \frac{\Delta \; {R\left( {x_{i},ɛ,f} \right)}}{G\left( {x_{i},{ɛ = 0},{f = f_{0}}} \right)}} & \left( {{Eq}.\mspace{14mu} 40} \right) \end{matrix}$

where both the initial contour location and the gradient are determined from the resist image at NC, i.e. R₀(x,y). The 2-dimensional edge shift can be calculated separately in x and y direction by the partial image derivative in each direction, or as an absolute shift using an absolute gradient value, i.e. the geometrical sum of S_(x)=R₀(x,y)−R₀(x−1,y) and S_(y)=R₀(x,y)−R₀(x,y−1), i.e., the absolute gradient value s=√{square root over (S_(x) ²S_(y) ²)}.

From the foregoing explanation, the edge shift can be directly expressed as a function of the differential images defined above:

$\begin{matrix} {{\Delta \; {{EP}\left( {x_{i},ɛ,f} \right)}} = {\frac{1}{S\left( x_{i} \right)}\left\lbrack {{{ɛ \cdot P}\left\{ {I_{0}\left( x_{i} \right)} \right\}} + {{\left( {1 + ɛ} \right) \cdot f \cdot P}\left\{ {a\left( x_{i} \right)} \right\}} + {{\left( {1 + ɛ} \right) \cdot f^{2} \cdot P}\left\{ {b\left( x_{i} \right)} \right\}}} \right\rbrack}} & \left( {{Eq}.\mspace{14mu} 41} \right) \end{matrix}$

while changes in CD or line widths can be determined from adding the individual edge placement shifts on either side of a line, resulting generally in ΔCD=2·ΔEP. Clearly, (Eq.41) will be able to reproduce the typical 2^(nd) order-like through-focus behavior of CD or EPE curves. More importantly, after the set of images such as [R₀, P{I₀}, P{a}, P{b}] has been calculated, which may be accomplished with only ˜1× more computation than simulating the single image at NC (assuming that fewer TCC terms are required for sufficient accuracy on the differentials), (Eq.41) may be applied to map out analytically the complete PW for every single edge position on a design, without the need for any further time-consuming image simulation. A generic flow diagram to illustrate this method is provided in FIG. 5.

Referring to FIG. 5, the first step (Step 80) entails defining the process specific parameters associated with the lithography process and system that will be utilized in the imaging process. Thereafter, derivative TCCs A and B are generated utilizing (Eq.14) (Step 82). In Step 84, calibration test data is acquired for multiple process window conditions. In Step 85, model parameters for R_(O){ } and/or associated filter P{ } are determined utilizing in part the results of Step 82. Next, the target mask pattern or design is defined (Step 86). The process then proceeds to generate images such as R_(O)(x), P{I_(O)}, P{a} and P{b} in Step 88. Next, the simulated image is synthesized, NC contours are extracted, and feature EPEs are determined at a given set of edge positions {x_(i)} (Step 90).

The process then proceeds to Step 92 to determine EPE or CD variations through process window at edge positions {x_(i)}. Finally, in Step 94, the results obtained in Step 92 are analyzed to determine whether the resulting image is within a predefined error tolerance, thus, determining a common process window as well as identifying any problem area (i.e., hot-spots) within the design.

The methods detailed above, and in particular (Eq.41) can be applied very flexibly for a wide range of tasks in lithography design inspection. Some of these applications are briefly outlined below. However, it is noted that the present invention is not limited to the applications disclosed herein.

For any particular edge or CD, (Eq.41) allows straightforward determination of the focus latitude (=DOF (Depth of Focus)) at nominal dose, for a given tolerance of CD, EP or line end variation.

For any particular edge or CD, (Eq.41) allows straightforward determination of the exposure dose at nominal focus, for a given tolerance of CD, EP or line end variation.

For any particular edge or CD, (Eq.41) allows straightforward mapping of the shape, center and area of the PW in {F,E} space or a generalized PW space, for a given tolerance of CD, EP or line end variation.

For a set of edges or CDs covering the full chip design and all relevant pattern/feature types, the common process window of the design can be efficiently calculated, and process corrections may be derived in order to center the common PW.

Critical, limiting patterns may be identified that define the inner boundaries of the common PW, by either having off-centered PWs or small PWs.

The common PW area may be mapped out as a function of tolerance specs on EP or CD variations. This sensitivity analysis may provide a yield estimate depending on design sensitivity.

Design hot spots may be identified from a full-chip analysis using (Eq.41), as patterns with PW area, DOF or exposure latitude falling below a certain threshold. The behavior of these critical patterns may then be investigated in detail by full-PW simulations, i.e. using the full simulation model for repeated image and resist contour simulation at many points across the PW.

Application in Calibration

Typically, the amount of optical tuning in calibration is quite small since the information on optical parameters is usually quite accurate. As a result, the dependence on all the optical parameters can be accurately expanded into a polynomial of the relevant parameters, for example, the aerial image can be expressed as a polynomial of optical parameters such as focus and NA, as shown in (Eq.20).

Therefore, such polynomial expansion can be applied in calibration in two aspects. First, the expansion can be applied to speed up the calibration process. In current calibration processes, the optical and resist parameters are typically identified through the wafer CDs of certain known patterns. For a complete set of optical parameters such as NA, focus f, dose E, and etc, we can determine the TCC as a complicated function of these parameters, i.e.,

TCC=TCC(NA,f,ε, . . . )  (Eq.42)

Given a pattern, denoted by its mask image M(k) (in frequency domain), the aerial image I(x) can then be computed (Eq.1) or (Eq.3). Next, the resist image R(x) can be evaluated by (Eq.21) and resist contour and CD can be subsequently determined. This process is demonstrated as steps 32, 34, and 36 in FIG. 2. For notational simplification, this process is encapsulated in a function SIMU(•), i.e.,

CD=SIMU(M,p ₁ , . . . , p _(N))  (Eq.43)

where M is the mask image, and p₁, . . . , p_(N) are N model parameters which may include the optical parameters such as NA, f, ε, . . . , and the resist parameters. Although CDs have been used in this description as an exemplary calibration target to demonstrate the methodology, the methodology can be applied to any other calibration target, such as contours and so on.

The calibration process is essentially an inverse problem of model simulation, i.e., in calibration, given L Mask images (M₁, M₂, . . . , M_(L)) and corresponding measured wafer CDs (CD₁, CD₂, . . . , CD_(L)), it is desired that the optical and resist parameters be identified. In practice, the optimal parameter values correspond to

$\begin{matrix} {\underset{p_{1},\ldots,p_{N}}{\arg \mspace{14mu} \min}{\sum\limits_{l = 1}^{L}\; \left( {{{SIMU}\left( {M_{l},p_{1},\ldots,p_{N}} \right)} - {CD}_{l}} \right)^{2}}} & \left( {{Eq}.\mspace{14mu} 44} \right) \end{matrix}$

That is, the optimal parameter values minimize the object function

${\sum\limits_{l = 1}^{L}\; \left( {{{SIMU}\left( {M_{l},p_{1},\ldots,p_{N}} \right)} - {CD}_{l}} \right)^{2}},$

which is the Euclidean distance between measured and simulated wafer CDs. Brute-force search is used in current calibrations, especially for optical parameter calibrations. With reference to FIG. 8, in Brute-force search, a possible search range of the parameters is first determined at step 800. For example, the search range for each parameter p_(n) (n=1, . . . , N) is typically a finite set of points: {p_(n,1), . . . , p_(n,L) _(n) }. Consequently, there are a total of N such sets for N parameters, and the total number of possible parameter combinations (possible process conditions) is

${SR} = {\prod\limits_{n = 1}^{N}\; {L_{n}.}}$

Then for each combination of parameter values (that is, for each possible process condition), the corresponding optical and resist models are computed at step 802. For various mask patterns 803, contours and CDs may be predicted at step 804. Predicted contours and CDs may be compared at step 806 with corresponding measured contours and CDs 805. After exhausting all possible parameter values combinations (i.e., possible process conditions), an optimal one with the minimum Euclidean distance is chosen at step 808. As the SIMU(•) function is complicated, the computation complexity of such brute-force approach often becomes prohibitively high when the total number of possible parameter combinations (SR) becomes large. For example, if the number of searching grid points for each parameter is a constant L (i.e., L_(n)=L for any n=1, . . . , N), then the computational complexity is exponential with respect to N, the number of parameters to be searched.

Now, with reference to FIG. 9, in the described polynomial expansion approach, the SIMU(•) function can be approximated by a simple polynomial of the parameters (e.g., replacing (Eq.43) by (Eq.41) if only focus and dose are to be calibrated), then the computational efficiency is vastly improved. Therefore, in one embodiment, one possible process condition is first chosen as a “nominal condition” at step 902 and the “nominal” CDs is computed in a traditional way at step 904. Then polynomial expansion can be applied with respect to the “nominal condition” and, at step 906, CDs in all other process condition can be computed using (Eq.41). For example, if only focus and dose are being calibrated and f₀ and ε₀ denote the “nominal” focus and dose, respectively, (Eq.44) becomes:

$\begin{matrix} {\underset{f,ɛ}{\arg \mspace{14mu} \min}{\sum\limits_{l = 1}^{L}\; \left( {{{SIMU}\left( {M_{l},f_{0},ɛ_{0}} \right)} + {\frac{2}{S\left( x_{l} \right)}\left\lbrack {{{\left( {{ɛ\text{/}ɛ_{0}} - 1} \right) \cdot P}\left\{ {I_{0}\left( x_{l} \right)} \right\}} + {ɛ\text{/}{ɛ_{0} \cdot \left( {f - f_{0}} \right) \cdot P}\left\{ {a\left( x_{l} \right)} \right\}} + {ɛ\text{/}{ɛ_{0} \cdot \left( {f - f_{0}} \right)^{2} \cdot P}\left\{ {b\left( x_{l} \right)} \right\}}} \right\rbrack} - {CD}_{l}} \right)^{2}}} & \left( {{Eq}.\mspace{14mu} 45} \right) \end{matrix}$

Note that in this embodiment, for each mask image M₁,SIMU( ) function is only evaluated once. That is, for all possible process conditions, SIMU( ) is only evaluated for the “nominal condition”, which significantly reduces the computational cost.

This polynomial expansion has a further application in calibration. The model is linear with respect to certain parameters, such as NA. For such linear terms, the calibration can be further improved in another embodiment. If the CD is linear with respect to all the parameters, then for mask image M, the SIMU(•) function (Eq.43) can be rewritten as

$\begin{matrix} {{{CD} = {{{SIMU}\left( {M,p_{10},\ldots,p_{N\; 0}} \right)} + {\sum\limits_{n = 1}^{N}\; {c_{n} \cdot \left( {p_{n} - p_{n\; 0}} \right)}}}}{{{{where}\mspace{14mu} {c_{n}(M)}} = \left. \frac{\partial{{SIMU}\left( {M,p_{1},\ldots,p_{N}} \right)}}{\partial p_{n}} \right|_{{p_{1} = p_{10}},\ldots,{p_{N} = p_{N\; 0}}}},}} & \left( {{Eq}.\mspace{14mu} 46} \right) \end{matrix}$

(n=1, . . . , N) are constant coefficients computed using the previous formula and p_(no) is the “nominal” value of p_(n). Note that c_(n)(M) are independent of all the parameters p, (n=1, . . . , N) but depend on the mask image.

As a result, (Eq.44) can be rewritten as

$\begin{matrix} {\underset{p_{1},\ldots,p_{N}}{\arg \mspace{14mu} \min}{\sum\limits_{l = 1}^{L}\; \left( {{{SIMU}\left( {M_{l},p_{10},\ldots,p_{N\; 0}} \right)} + {\sum\limits_{n = 1}^{N}\; {{c_{n}\left( M_{l} \right)} \cdot \left( {p_{n} - p_{n\; 0}} \right)}} - {CD}_{l}} \right)^{2}}} & \left( {{Eq}.\mspace{14mu} 47} \right) \end{matrix}$

(Eq.47) can again be solved using the method of least-square, which leads to a global optimal solution. Derivatives of the Euclidean distance with respect to each parameter p_(m) (m=1, . . . , N) are computed and set to 0, i.e.,

$\begin{matrix} {{\sum\limits_{l = 1}^{L}\; {{c_{m}\left( M_{l} \right)}\left( {{{SIMU}\left( {M_{l},p_{10},\ldots,p_{N\; 0}} \right)} + {\sum\limits_{n = 1}^{N}\; {{c_{n}\left( M_{l} \right)} \cdot \left( {p_{n} - p_{n\; 0}} \right)}} - {CD}_{l}} \right)}} = 0} & \left( {{Eq}.\mspace{14mu} 48} \right) \end{matrix}$

There are N (m=1, . . . , N) such linear equations of the N unknown parameters p_(n) (n=1, . . . , N), and the solution to these equations represents the sought global optimum.

The general formula for the solution is:

$\begin{matrix} {\begin{bmatrix} {\hat{p}}_{1} \\ {\hat{p}}_{2} \\ M \\ {\hat{p}}_{N} \end{bmatrix} = {\begin{bmatrix} p_{10} \\ p_{20} \\ M \\ p_{N\; 10} \end{bmatrix} + {\Delta^{- 1}\Gamma}}} & \left( {{Eq}.\mspace{14mu} 49} \right) \end{matrix}$

where Δ is a N×N matrix defined as

$\Delta = \begin{bmatrix} {\sum\limits_{l = 1}^{L}\; {{c_{1}\left( M_{l} \right)}{c_{1}\left( M_{l} \right)}}} & {\sum\limits_{l = 1}^{L}\; {{c_{1}\left( M_{l} \right)}{c_{2}\left( M_{l} \right)}}} & \Lambda & {\sum\limits_{l = 1}^{L}\; {{c_{1}\left( M_{l} \right)}{c_{N}\left( M_{l} \right)}}} \\ {\sum\limits_{l = 1}^{L}\; {{c_{2}\left( M_{l} \right)}{c_{1}\left( M_{l} \right)}}} & {\sum\limits_{l = 1}^{L}\; {{c_{2}\left( M_{l} \right)}{c_{2}\left( M_{l} \right)}}} & \Lambda & {\sum\limits_{l = 1}^{L}\; {{c_{2}\left( M_{l} \right)}{c_{N}\left( M_{l} \right)}}} \\ M & M & O & M \\ {\sum\limits_{l = 1}^{L}\; {{c_{N}\left( M_{l} \right)}{c_{1}\left( M_{l} \right)}}} & {\sum\limits_{l = 1}^{L}\; {{c_{N}\left( M_{l} \right)}{c_{2}\left( M_{l} \right)}}} & \Lambda & {\sum\limits_{l = 1}^{L}\; {{c_{N}\left( M_{l} \right)}{c_{N}\left( M_{l} \right)}}} \end{bmatrix}$

and Γ is an N-dimensional vector defined as

$\Gamma = \begin{bmatrix} {\sum\limits_{l = 1}^{L}\; {{c_{1}\left( M_{l} \right)}\left( {{CD}_{l} - {{SIMU}\left( {M_{l},p_{10},\ldots,p_{N\; 0}} \right)}} \right)}} \\ {\sum\limits_{l = 1}^{L}\; {{c_{2}\left( M_{l} \right)}\left( {{CD}_{l} - {{SIMU}\left( {M_{l},p_{10},\ldots,p_{N\; 0}} \right)}} \right)}} \\ M \\ {\sum\limits_{l = 1}^{L}\; {{c_{N}\left( M_{l} \right)}\left( {{CD}_{l} - {{SIMU}\left( {M_{l},p_{10},\ldots,p_{N\; 0}} \right)}} \right)}} \end{bmatrix}$

This approach removes the need to determine the search range of the parameter and to simulate the resist CDs over many possible process conditions approach. Thus both speed and accuracy are improved.

Further, for those non-linear parameters (including all linear parameters), a number of numerical optimization algorithms can be employed if a good initial estimate of the parameters is available. Newton's method (also known as Newton-Raphson method, Newton-Fourier method, or multi-stage linearized least square fitting) may be initially used as an example to demonstrate how to solve it. If each parameter p_(n) (n=1, . . . , N) is within in a small range around its “nominal” value p_(n0), then function SIMU( )can be expanded around p_(n0) (n=1, . . . , N) as shown in (Eq.46). Note that all higher-order effects are ignored here so that the optimization problem can be treated as a least square fitting problem and solved using (Eq.48). Denoting the solution to (Eq.48) using {circumflex over (p)}_(n) (n=1, . . . , N), and they represent a better estimation of the parameters. p_(n0) (n=1, . . . , N) can then be replaced by {circumflex over (p)}_(n) (n=1, . . . , N) and the polynomial expansion can be repeated, this time around {circumflex over (p)}_(n) (n=1, . . . , N). This process of least square fitting and linear expansion can then be repeated until the parameter values converge or until some iteration constraint is encountered, such as maximum number of iterations. Note that 2Δ defined in (Eq.49) is also called the Hessian matrix of the object function

$\sum\limits_{l = 1}^{L}\; \left( {{{SIMU}\left( {M_{l},p_{1},\ldots,p_{N}} \right)} - {CD}_{l}} \right)^{2}$

and 2Γ is in fact the gradient of the object function. This process is illustrated in FIG. 10.

It will be appreciated that this problem can be categorized within the general field of multi-dimensional optimization. Therefore, a number of alternative methods can be employed, including, for example, Gradient descent algorithm, Gaussian-Newton algorithm, Levenberg-Marquardt algorithm, etc. All of these methods can eliminate the requirement of the search range and can improve the speed and calibration accuracy significantly.

Further, there may exist certain parameters, such as focus, which have a strong second or higher order effect on CD. For such parameters, the linearized approximation (Eq.46) may not work well, as it may require many iterations to converge. In this case, higher order terms may be retained, e.g., using (Eq.20). To solve (Eq.44) with these terms, the method of linear least square directly may not be applied. However, Newton's method, Gradient descent algorithm, Gaussian-Newton algorithm, Levenberg-Marquardt algorithm, etc. can be employed to compute optimal parameters, since these algorithms work for general non-linear optimization problems. Although such an approach may increase the complexity of each iteration, a better approximation for the CD may be attained with higher order terms and thereby can reduce the number of iterations.

In most applications, there is ample knowledge of the optical parameters. However, it may be difficult to obtain a good initial estimate on the resist parameters for certain resist model forms; and resist parameter variations may not be confined to a small range around the initial estimate. In such cases, an alternative optimization algorithm may be employed whereby optical parameters may be fixed and, subsequently, brute search or non-linear optimization methods such as simplex algorithm may be used to optimize resist parameters. Resist parameters may then be fixed and multi-dimensional optimization algorithms can be used to optimize optical parameters. These iterative optimizations can be repeated until convergence.

FIG. 6 is a block diagram that illustrates a computer system 100 which can assist in the simulation method disclosed herein. Computer system 100 includes a bus 102 or other communication mechanism for communicating information, and a processor 104 coupled with bus 102 for processing information. Computer system 100 also includes a main memory 106, such as a random access memory (RAM) or other dynamic storage device, coupled to bus 102 for storing information and instructions to be executed by processor 104 and co-processor 105. Main memory 106 also may be used for storing temporary variables or other intermediate information during execution of instructions to be executed by processor 104. Computer system 100 further includes a read only memory (ROM) 108 or other static storage device coupled to bus 102 for storing static information and instructions for processor 104. A storage device 110, such as a magnetic disk or optical disk, is provided and coupled to bus 102 for storing information and instructions.

Computer system 100 may be coupled via bus 102 to a display 112, such as a cathode ray tube (CRT) or flat panel or touch panel display for displaying information to a computer user. An input device 114, including alphanumeric and other keys, is coupled to bus 102 for communicating information and command selections to processor 104. Another type of user input device is cursor control 116, such as a mouse, a trackball, or cursor direction keys for communicating direction information and command selections to processor 104 and for controlling cursor movement on display 112. This input device typically has two degrees of freedom in two axes, a first axis (e.g., x) and a second axis (e.g., y), that allows the device to specify positions in a plane. A touch panel (screen) display may also be used as an input device.

In certain embodiments, portions of the simulation process may be performed by computer system 100 in response to processor 104 executing one or more sequences of one or more instructions contained in main memory 106. Such instructions may be read into main memory 106 from another computer-readable medium, such as storage device 110. Execution of the sequences of instructions contained in main memory 106 causes processor 104 to perform the process steps described herein. One or more processors in a multi-processing arrangement may also be employed to execute the sequences of instructions contained in main memory 106. In certain embodiments, hard-wired circuitry may be used in place of or in combination with software instructions to implement the invention. Thus, embodiments of the invention are not limited to any specific combination of hardware circuitry and software.

The term “computer-readable medium” as used herein refers to any medium that participates in providing instructions to processor 104 for execution. Such a medium may take many forms, including but not limited to, non-volatile media, volatile media, and transmission media. Non-volatile media include, for example, optical or magnetic disks, such as storage device 110. Volatile media include dynamic memory, such as main memory 106. Transmission media include coaxial cables, copper wire and fiber optics, including the wires that comprise bus 102. Transmission media can also take the form of acoustic or electromagnetic waves, including visible light, radio frequency (RF) and infrared (IR) used for data communications. Common forms of computer-readable media include, for example, a floppy disk, a flexible disk, hard disk, magnetic tape, any other magnetic medium, a CD-ROM, DVD, Blu-Ray any other optical medium, punch cards, paper tape, any other physical medium with patterns of holes, a RAM, a PROM, and EPROM, a FLASH-EPROM, any other memory chip or cartridge, a carrier wave as described hereinafter, or any other medium from which a computer can read.

Various forms of computer readable media may be involved in carrying one or more sequences of one or more instructions to processor 104 for execution. For example, the instructions may initially be borne on a magnetic disk of a remote computer. The remote computer can load the instructions into its dynamic memory and send the instructions over a telephone line using a modem. A modem local to computer system 100 can receive the data on the telephone line and use an infrared transmitter to convert the data to an infrared signal. An infrared detector coupled to bus 102 can receive the data carried in the infrared signal and place the data on bus 102. Bus 102 carries the data to main memory 106, from which processor 104 retrieves and executes the instructions. The instructions received by main memory 106 may optionally be stored on storage device 110 either before or after execution by processor 104.

Computer system 100 may also include a communication interface 118 coupled to bus 102. Communication interface 118 provides a two-way data communication coupling to a network link 120 that is connected to a local network 122. For example, communication interface 118 may be an integrated services digital network (ISDN) card or a modem to provide a data communication connection to a corresponding type of telephone line. As another example, communication interface 118 may be a local area network (LAN) card to provide a data communication connection to a compatible LAN. Wireless links may also be implemented. In any such implementation, communication interface 118 sends and receives electrical, electromagnetic or optical signals that carry digital data streams representing various types of information.

Network link 120 typically provides data communication through one or more networks to other data devices. For example, network link 120 may provide a connection through local network 122 to a host computer 124 or to data equipment operated by an Internet Service Provider (ISP) 126. ISP 126 in turn provides data communication services through the worldwide packet data communication network, now commonly referred to as the “Internet” 128. Local network 122 and Internet 128 both use electrical, electromagnetic or optical signals that carry digital data streams. The signals through the various networks and the signals on network link 120 and through communication interface 118, which carry the digital data to and from computer system 100, are exemplary forms of carrier waves transporting the information.

Computer system 100 can send messages and receive data, including program code, through the network(s), network link 120, and communication interface 118. In the Internet example, a server 130 might transmit a requested code for an application program through Internet 128, ISP 126, local network 122 and communication interface 118. According to certain aspects of the invention a downloaded application can provide for the illumination optimization of the embodiment, for example. The received code may be executed by processor 104 as it is received, and/or stored in storage device 110, or other non-volatile storage for later execution. In this manner, computer system 100 may obtain application code in the form of a carrier wave.

FIG. 7 schematically depicts an example of lithographic projection apparatus whose performance could be simulated utilizing the processes described herein. The apparatus in the example comprises:

-   -   a radiation system IL, for supplying a projection beam B of         radiation. In this particular case, the radiation system also         comprises a radiation source LA;     -   a first object table (mask table) MT provided with a mask holder         for holding a mask MA (e.g., a reticle), and connected to first         positioning means for accurately positioning the mask with         respect to item PS;     -   a second object table (substrate table) WT provided with a         substrate holder for holding a substrate W (e.g., a         resist-coated silicon wafer), and connected to second         positioning means for accurately positioning the substrate with         respect to item PS;     -   a projection system (“lens”) PS (e.g., a refractive, catoptric         or catadioptric optical system) for imaging an irradiated         portion of the mask MA onto a target portion C (e.g., comprising         one or more dies) of the substrate W.

As depicted herein, the apparatus is of a transmissive type (i.e., has a transmissive mask). However, in general, it may also be of a reflective type, for example (with a reflective mask). Alternatively, the apparatus may employ another kind of patterning means as an alternative to the use of a mask; examples include a programmable mirror array or LCD matrix.

The source LA (e.g., a mercury lamp or excimer laser) produces a beam of radiation. This beam is fed into an illumination system (illuminator) IL, either directly or after having traversed conditioning means, such as a beam expander or beam delivery system BD, for example. The illuminator IL may comprise adjusting means AD for setting the outer and/or inner radial extent (commonly referred to as σ-outer and σ-inner, respectively) of the intensity distribution in the beam. In addition, it will generally comprise various other components, such as an integrator IN and a condenser CO. In this way, the beam B impinging on the mask MA has a desired uniformity and intensity distribution in its cross-section.

With regard to FIG. 7, the source LA may be within the housing of the lithographic projection apparatus (as is often the case when the source LA is a mercury lamp, for example), but that it may also be remote from the lithographic projection apparatus, the radiation beam that it produces being led into the apparatus (e.g., with the aid of suitable directing mirrors); this latter scenario is often the case when the source LA is an excimer laser (e.g., based on KrF, ArF or F₂ lasing). Various embodiments of the invention encompass at least both of these scenarios.

The beam B subsequently intercepts the mask MA, which is held on a mask table MT. Having traversed the mask MA, the beam PB passes through the lens PS, which focuses the beam B onto a target portion C of the substrate W. With the aid of the second positioning means (and interferometric measuring means IF), the substrate table WT can be moved accurately, e.g. so as to position different target portions C in the path of the beam B. Similarly, the first positioning means can be used to accurately position the mask MA with respect to the path of the beam B, e.g., after mechanical retrieval of the mask MA from a mask library, or during a scan. In general, movement of the object tables MT, WT will be realized with the aid of a long-stroke module (coarse positioning) and a short-stroke module (fine positioning), which are not explicitly depicted in FIG. 7. However, in the case of a wafer stepper (as opposed to a step-and-scan tool) the mask table MT may just be connected to a short stroke actuator, or may be fixed. Patterning device MA and substrate W may be aligned using alignment mark M1, M2 in the patterning device, and alignment mark P1, P2 on the wafer, as required.

The depicted tool can be used in different modes including:

-   -   In step mode, the mask table MT is kept essentially stationary,         and an entire mask image is projected in one go (i.e., a single         “flash”) onto a target portion C. The substrate table WT is then         shifted in the x and/or y directions so that a different target         portion C can be irradiated by the beam B;     -   In scan mode, essentially the same scenario applies, except that         a given target portion C is not exposed in a single “flash”.         Instead, the mask table MT is movable in a given direction (the         so-called “scan direction”, e.g., the y direction) with a speed         v, so that the projection beam B is caused to scan over a mask         image; concurrently, the substrate table WT is simultaneously         moved in the same or opposite direction at a speed V=Mv, in         which M is the magnification of the lens PS (typically, M=1/4 or         1/5). In this manner, a relatively large target portion C can be         exposed, without having to compromise on resolution.

The concepts disclosed herein may simulate or mathematically model any generic imaging system for imaging sub wavelength features, and may be especially useful with emerging imaging technologies capable of producing wavelengths of an increasingly smaller size. Emerging technologies already in use include EUV (extreme ultra violet) lithography that is capable of producing a 193 nm wavelength with the use of an ArF laser, and even a 157 nm wavelength with the use of a Fluorine laser. Moreover, EUV lithography is capable of producing wavelengths within a range of 20-5 nm by using a synchrotron or by hitting a material (either solid or a plasma) with high energy electrons in order to produce photons within this range. Because most materials are absorptive within this range, illumination may be produced by reflective mirrors with a multi-stack of Molybdenum and Silicon. The multi-stack mirror has a 40 layer pairs of Molybdenum and Silicon where the thickness of each layer is a quarter wavelength. Even smaller wavelengths may be produced with X-ray lithography. Typically, a synchrotron is used to produce an X-ray wavelength. Since most material is absorptive at x-ray wavelengths, a thin piece of absorbing material defines where features would print (positive resist) or not print (negative resist).

While the concepts disclosed herein may be used for imaging on a substrate such as a silicon wafer, it shall be understood that the disclosed concepts may be used with any type of lithographic imaging systems, e.g., those used for imaging on substrates other than silicon wafers.

Additional Descriptions of Certain Aspects of the Invention

The foregoing descriptions of the invention are intended to be illustrative and not limiting. For example, those skilled in the art will appreciate that the invention can be practiced with various combinations of the functionalities and capabilities described above, and can include fewer or additional components than described above. Certain additional aspects and features of the invention are further set forth below, and can be obtained using the functionalities and components described in more detail above, as will be appreciated by those skilled in the art after being taught by the present disclosure.

Certain embodiments of the invention provide methods for calibrating a photolithographic system. Some of these embodiments comprise the steps of obtaining a plurality of measured dimensions of circuit patterns generated using a configuration of a photolithographic process, generating a plurality of estimated dimensions of the circuit patterns using a model of the configuration of the photolithographic process, for certain of the circuit patterns, calculating a polynomial fit between the estimated dimensions and predefined parameters related to the configuration and calibrating the photolithographic process based on the polynomial fit, wherein calibrating the photolithographic process includes the step of minimizing differences between the estimated and measured dimensions using an optimizing algorithm. In some of these embodiments, the estimated and measured dimensions include dimensions defining contour shapes of the circuit patterns. In some of these embodiments, the step of minimizing differences includes minimizing the sum of squared differences between the estimated and measured dimensions. In some of these embodiments, certain of the estimated and measured dimensions relate to critical dimensions of the circuit patterns. In some of these embodiments, calibrating the photolithographic process includes determining a linear relationship between at least one optical parameter and one or more estimated critical dimensions. In some of these embodiments, calibrating the photolithographic process includes determining a linear relationship between at least one optical parameter and the plurality of estimated dimensions and a second or higher order relationship between at least one optical parameter and one or more estimated critical dimensions.

In some of these embodiments, calculating a polynomial fit includes performing a least square optimization algorithm on estimated values of the one or more critical dimensions for each of the at least one optical parameter. In some of these embodiments, the least square optimization algorithm is a gradient descent algorithm. In some of these embodiments, the least square optimization algorithm is a Gaussian-Newton algorithm. In some of these embodiments, the least square optimization algorithm is a Levenberg-Marquardt algorithm. In some of these embodiments, calibrating the photolithographic process includes computing the estimated dimensions using a fitted polynomial of selected parameters. In some of these embodiments, calculating the polynomial fit of selected parameters further includes using differential transmission cross coefficients. In some of these embodiments, calculating the polynomial fit includes performing a least square approximation for each one of the plurality of estimated dimensions using polynomials of certain model parameters.

In some of these embodiments, the at least one optical parameter is fixed and wherein calibrating the photolithographic process includes optimizing a non-linear resist threshold setting. In some of these embodiments, the at least one optical parameter includes a defocus setting. In some of these embodiments, the at least one optical parameter includes an exposure dose setting. In some of these embodiments, the at least one optical parameter includes a focus setting and a numerical aperture setting. In some of these embodiments, calculating the polynomial fit includes expressing the critical dimensions as a polynomial of the focus setting. In some of these embodiments, calculating the polynomial fit includes generating an aerial image using a polynomial of the focus setting and the numerical aperture setting. In some of these embodiments, calculating the polynomial fit includes expressing a resist image as a polynomial of the focus setting and the numerical aperture setting. In some of these embodiments, calculating includes approximating a focus-related variation in image intensity as a second-order polynomial of the focus setting for each of a plurality of image points. In some of these embodiments, the polynomial fit is a second order fit.

In some of these embodiments, providing the plurality of estimated dimensions includes obtaining a plurality of nominal critical dimensions calculated for a nominal process condition, the nominal process condition characterizing a nominal photolithographic process and computing the plurality of estimated dimensions using the polynomial fit and the plurality of nominal critical dimensions. In some of these embodiments, calculating the polynomial fit includes using pre-computed differential transmission cross coefficients associated with the configuration. In some of these embodiments, calculating the polynomial fit includes providing a polynomial series expansion of an aerial image as a function of the at least one optical parameter. In some of these embodiments, the aerial image is calculated based on pre-computed transmission cross coefficients. In some of these embodiments, calculating the polynomial fit includes providing a polynomial series expansions of a derivative of an aerial image as a function of the at least one optical parameter. In some of these embodiments, the aerial image is calculated based on derivatives of pre-computed transmission cross coefficients. One or more of the above described methods may be embodied in computer readable medium in the form of instructions and/or computer programs which, when executed, cause a computer to perform steps of the method.

In an embodiment of the invention there is provided a calibration method, comprising generating a model for a lithographic process, the model comprising a polynomial series expansion around a nominal value of a physical parameter of the lithographic process. The calibration method further comprises fitting the model to measured dimensions of imaging results obtained by applying the lithographic process at a plurality of values of the physical parameter by optimizing at least one polynomial expansion coefficient. Note that by applying a polynomial series expansion, a separation of the model into an optical model part and a resist model part is not required: the polynomial series expansion can be made for the integrated (i.e. combined) model. Additionally the calibration is reduces to a linear problem even if the parameters are non-linear.

In an embodiment the calibration method, the calibration method comprises optimizing at least two polynomial expansion coefficients. By having at least two in stead of one polynomial expansion coefficient, the accuracy of the model is improved.

In a yet further embodiment, the calibration method comprises using weight factors for different values of the physical parameter. This is because some values of the physical parameters are less likely then other combinations.

In a further embodiment the calibration method comprises a further polynomial series expansion around a further nominal value of a further physical parameter of the lithographic process. The imaging results are obtained at a plurality of values of the further physical parameter and the fitting of the model comprises optimizing at least one further polynomial expansion coefficient corresponding to the further polynomial series expansion. By using two polynomial series expansions for the physical parameter and the further physical parameter, the equations to be solved are linearized enabling an efficient way to solve them. Because of this use of two polynomial series expansions, the fits to the model do not rely on the combinations of the values of the physical parameters (i.e. do not pose restrictions to the combinations) used for applying the lithographic process.

In a yet further embodiment, the calibration method comprises using weight factors for different combinations of the values of the physical parameter and the further physical parameter. This is because some combinations of different values of the physical parameter and the further physical parameter are less likely then other combinations.

In an embodiment, the model is used to explore the so called process latitude, i.e. the robustness of the lithographic process against unwanted variations of the physical parameter(s). By applying the method of the embodiment, the model is calibrated in an efficient way and equally efficiently. Then it is be explored how for instance the CD-uniformity or the aerial image intensity varies with such unwanted variations of the values of the physical parameter. If this is within acceptable limits, the nominal values are accepted and the lithographic process is applied to exposure substrates to a patterned beam of radiation. In a further embodiment, the steps of the calibration method above are repeated for different combinations of the nominal values of the physical parameter and the further physical parameter. Then the robustness for the different combinations is checked and the most robust combination is chosen for application of the lithographic process. Other criteria for choosing an optimal combination of parameter values are possible (for instance low dose and acceptable process latitude as a low dose may enable a larger process speed). It will be clear to the skilled person that this embodiment can also be applied to variations in the values of just the physical parameter in an analogue way. Of course, the values for the physical parameter and the further physical parameter obtained by an embodiment of the invention can then be used as values for parameters when actually applying the actual lithographic process comprising exposing a substrate to a patterned beam of radiation.

Although the present invention has been described and illustrated in detail, it is to be clearly understood that the same is by way of illustration and example only and is not to be taken by way of limitation, the scope of the present invention being limited only by the terms of the appended claims. 

We claim:
 1. A calibration method implemented by a computer, the method comprising: generating, using the computer, a model for a lithographic process, the model comprising a polynomial series expansion around a nominal value of a physical parameter of the lithographic process; and fitting the model to measured dimensions of imaging results obtained by applying the lithographic process at a plurality of values of the physical parameter by optimizing at least one polynomial expansion coefficient.
 2. The calibration method of claim 1, further comprising: optimizing at least two polynomial expansion coefficients.
 3. The calibration method according to claim 1, further comprising: using weight factors for different values of the physical parameter.
 4. The calibration method according to claim 1, further comprising: generating differential transmission cross coefficients in a part of the model corresponding to an imaging process.
 5. The calibration method according to claim 1, further comprising: performing a further polynomial series expansion around a further nominal value of a further physical parameter of the lithographic process, wherein the imaging results are obtained at a plurality of values of the further physical parameter, and wherein fitting the model comprises optimizing at least one further polynomial expansion coefficient corresponding to the further polynomial series expansion.
 6. The calibration method according to claim 5, comprising: using weight factors for combinations of different values of the physical parameter and the further physical parameter.
 7. The calibration method according to claim 1, further comprising: determining a variation of a metric with variation of the parameter around the value of the nominal parameter using the model; repeating the generating and fitting steps at a different value of the nominal parameter; determining a further variation of a metric with variation of the parameter around the different value of the nominal parameter using the model; and selecting a value for the physical parameter based on the variation of the value of the physical parameter and the further variation of the value of the physical parameter.
 8. The calibration method of claim 1, wherein the physical parameter is related to one or more of an optical parameter and a resist parameter.
 9. The calibration method of claim 1, wherein the imaging result is associated with one or both of an aerial image and a resist image.
 10. The calibration method of claim 1, wherein the plurality of values of the physical parameter are selected within a process window.
 11. A non-transitory computer-readable storage medium having computer-executable instructions stored therein for causing a computer to perform a calibration method, the method comprising: generating, using the computer, a model for a lithographic process, the model comprising a polynomial series expansion around a nominal value of a physical parameter of the lithographic process; and fitting the model to measured dimensions of imaging results obtained by applying the lithographic process at a plurality of values of the physical parameter by optimizing at least one polynomial expansion coefficient.
 12. The computer-readable storage medium of claim 11, wherein the calibration method further comprises: optimizing at least two polynomial expansion coefficients.
 13. The computer-readable storage medium of claim 11, wherein the calibration method further comprises: using weight factors for different values of the physical parameter.
 14. The computer-readable storage medium of claim 11, wherein the calibration method further comprises: generating differential transmission cross coefficients in a part of the model corresponding to an imaging process.
 15. The computer-readable storage medium of claim 11, wherein the calibration method further comprises: performing a further polynomial series expansion around a further nominal value of a further physical parameter of the lithographic process, wherein the imaging results are obtained at a plurality of values of the further physical parameter, and wherein fitting the model comprises optimizing at least one further polynomial expansion coefficient corresponding to the further polynomial series expansion.
 16. The computer-readable storage medium of claim 15, wherein the calibration method further comprises: using weight factors for combinations of different values of the physical parameter and the further physical parameter.
 17. The computer-readable storage medium of claim 11, wherein the calibration method further comprises: determining a variation of a metric with variation of the parameter around the value of the nominal parameter using the model; repeating the generating and fitting steps at a different value of the nominal parameter; determining a further variation of a metric with variation of the parameter around the different value of the nominal parameter using the model; and selecting a value for the physical parameter based on the variation of the value of the physical parameter and the further variation of the value of the physical parameter.
 18. The computer-readable storage medium of claim 11, wherein the physical parameter is related to one or more of an optical parameter and a resist parameter.
 19. The computer-readable storage medium of claim 11, wherein the imaging result is associated with one or both of an aerial image and a resist image.
 20. The computer-readable storage medium of claim 11, wherein the plurality of values of the physical parameter are selected within a process window. 